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MATHEMATICAL

AND

PHYSICAL PAPERS

BY

Sir GEORGE GABRIEL STOKES, Bart.,

M.A., D.C.L., LL.D., F.R.S.,

FELLOW OF PEMBROKE COLLEGE AND LUCASIAN PROFESSOR OF MATHEMATICS
IN THE UNIVERSITY OF CAMBRIDGE.

Reprinted from the Original Journals and Transactions ,
with Additional Notes by the Author.

YOL. III.

CAMBRIDGE:

AT THE UNIVERSITY PRESS.

1901

3 ,

S

x 3

dTmniirtirgn

PRINTED BY J. AND C. E. OLAY,
AT THE UNIVERSITY PRESS.

PREFACE.

When the second volume of my collected papers was
published, it was my intention to have entered on some
rather elaborate and in part laborious calculations bearing on
two of the papers which appear in the present volume. These
were however put off from time to time in favour of other
matters which claimed my attention; but meanwhile time went
on, and I deeply regret to find how long it now is since the
second volume appeared. There are other papers which still
remain, and I hope, should life and health last, to put these
together without delay.

G. G. STOKES.

Malahide, Co. Dublin, September , 1901.

carnegie msnrurc

OF TEOiflOi-Q>»V L

CONTENTS.

PAGE

On the Effect of the Internal Friction of Fluids on the Motion of

Pendulums.1

Part I. Analytical Investigation.

Section I. Adaptation of the General Equations to the case of
the Fluid surrounding a Body which oscillates as a Pen¬
dulum. General Laws which follow from the form of the
Equations. Solution of the Equations in the case of an

Oscillating Plane.11

Section II. Solution of the Equations in the case of a Sphere
oscillating in a mass of Fluid either unlimited, or confined
by a Spherical Envelope concentric with the Sphere in its

position of Equilibrium.22

Section III. Solution of the Equations in the case of an Infinite
Cylinder oscillating in an unlimited mass of Fluid, in a

direction perpendicular to its axis.38

Section IV. Determination of the Motion of a Fluid about a
Sphere which moves uniformly with a small velocity. Justi¬
fication of the Application of the Solutions obtained in
Sections II. and III. to cases in which the extent of
Oscillation is not small in comparison with the radius
of the Sphere or Cylinder. Discussion of a difficulty which
presents itself with reference to the uniform motion of a

Cylinder in a Fluid.55

Section V. On the effect of Internal Friction in causing the
Motion of a Fluid to subside. Application to the case of

Oscillatory Waves.67

Part II. Comparison of Theory and Experiment.

Section I. Discussion of the Experiments of Baily, Bessel,

Coulomb, and Dubuat.76

Section II. Suggestions with reference to future Experiments . 123

An Examination of the possible effect of the Radiation of Heat on the

Propagation of Sound.142

CONTENTS.

*ii

PAGE

he Colours of Thick Plates. 155

Section I. Bings thrown on a Screen by a Concave Mirror consisting
of a lens dimmed at the first surface, and quicksilvered at the
back. Condition of distinctness when the Bings are thrown on
a Screen, or of fixity when they are viewed in Air. Investigation
of the phenomena observed when the Luminous Point is moved
in a direction perpendicular to the Axis of the Mirror . . 160

Section II. Bands formed by a Plane Mirror, and viewed directly

by the Eye.170

Section III. Rings formed by a Curved Mirror, and viewed directly
by the Eye, when the Luminous Point and its Image are not in

the same plane perpendicular to the axis.178

Section IV. Straight bands formed by a Plane Mirror at a consider¬
able angle of incidence, and viewed by the Eye, either directly, or

through a Telescope.184

Section V. On the Nature of the Deflection of the two Interfering

Streams from the course of the regularly Refracted Light . . 186

Section VI. Investigation of the Angles of Diffraction . . . 191

On a new Elliptic Analyser.197

On the Conduction of Heat in Crystals.203

On the Total Intensity of Interfering Light.228

On the Composition and Resolution of Streams of Polarized Light from

different Sources.283

Abstract of a paper “ On the Change of Refrangibility of Light ” . . 259

On the Change of Refrangibility of Light.267

Notes added during printing of the original paper .... 400

Special index for that paper.408

Addition, introduced in the reprint.410

Index. 415

*. to face p. 285

ERRATA IN VOL. II.

p. 334, equation (10). Insert “=” before “Jn/Li”

p. 355, equation (66), last term. For “ -245835” (log. = 1-390644) read
“•245270” (log. = 1-389644)

MATHEMATICAL AND PHYSICAL PAPERS.

[From the Transactions of the Cambridge Philosophical Society ,
Yol. ix. p. [8]].

On the Effect of the Inteenal Fbiction of Fluids on
the Motion of Pendulums.

[Read December 9, 1850.]

The great importance of the results obtained by means of the
pendulum has induced philosophers to devote so much attention
to the subject, and to perform the experiments with such a scrupu¬
lous regard to accuracy in every particular, that pendulum observa¬
tions may justly he ranked among those most distinguished by
modern exactness. It is unnecessary here to enumerate the
different methods which have been employed, and the several
corrections which must be made, in order to deduce from the
actual observations the result which would correspond to the ideal
case of a simple pendulum performing indefinitely small oscillations
in vacuum. There is only one of these corrections which bears on
the subject of the present paper, namely, the correction usually
termed the reduction to a vacuum . On account of the inconvenience
and expense attending experiments in a vacuum apparatus, the
observations are usually made in air, and it then becomes necessary
to apply a small correction, in order to reduce the observed result
to what would have been observed had the pendulum been swung
in a vacuum. The most obvious effect of the air consists in a
diminution of the moving force, and consequent increase in the
time of vibration, arising from the buoyancy of the fluid. The

1

S. III.

2 ON THE EFFECT OF THE INTERNAL FRICTION OF FLUIDS

correction for buoyancy is easily calculated from tbe first principles
of hydrostatics, and formed for a considerable time tlie^ only
correction which it was thought necessary to make for reduction to
a vacuum. But in the year 1828 Bessel, in a very important
memoir in which he determined by a new method the length
of the seconds’ pendulum, pointed out from theoretical considera¬
tions the necessity of taking account of the inertia of the air as
well as of its buoyancy. The numerical calculation of the effect of
the inertia forms a problem of hydrodynamics which Bessel did not
attack; but he concluded from general principles that a fluid, or
at any rate a fluid of small density, has no other effect on the time
of very small vibrations of a pendulum than that it diminishes its
gravity and increases its moment of inertia. In the case of a body
of which the dimensions are small compared with the length of the
suspending wire, Bessel represented the increase of inertia by that
of a mass equal to k times the mass of the fluid displaced, which
must be supposed'to be added to the inertia of the body itself.
This factor k he determined experimentally for a sphere a little
more than two inches in diameter, swung in air and in water.
The result for air, obtained in a rather indirect way, was k = 0*9459,
which value Bessel in a subsequent paper increased to 0*956. A
brass sphere of the above size having been swung in water with
two different lengths of wire in succession gave two values of k ,
differing a little from each other, and equal to only about two-
thirds of the value obtained for air.

The attention of the scientific world having been called to the
subject by the publication of Bessel’s memoir, fresh researches
both theoretical and experimental soon appeared. In order to
examine the effect of the air by a more direct method than that
employed by Bessel, a large vacuum apparatus was erected at the
expense of the Board of Longitude, and by means of this apparatus
Captain (now Colonel) Sabine, determined the effect of the air on
the time of vibration of a particular invariable pendulum. The
results of the experiments are contained in a memoir read before
the Royal Society in March 1829, and printed in the Philosophical
Transactions for that year. The mean of eight very consistent
experiments gave 1*655 as the factor by which for that pendulum
the old correction for buoyancy must be multiplied in order to
give the whole correction on account of the air. A very remark¬
able fact was discovered in the course of these experiments. While

ON THE MOTION OF PENDULUMS.

s

the effects of air at the atmospheric pressure and under a pressure
of about half an atmosphere were found to be as nearly as possible
proportional to the densities, it was found that the effect of hydro¬
gen at the atmospheric pressure was much greater, compared with
the effect of air, than corresponded with its density. In fact, it
appeared that the ratio of the effects of hydrogen and air on the
times of vibration was about 1 to 5|, while the ratio of the. densities
is only about 1 to 13. In speaking of this result Colonel Sabine
remarks, “ The difference of this ratio from that shewn by experi¬
ment is greater than can well be ascribed to accidental error in
the experiment, particularly as repetition produced results almost
identical. May it not indicate an inherent property in the elastic
fluids, analogous to that of viscidity in liquids, of resistance to the
motion of bodies passing through them, independently of their
density ? a property, in such case, possessed by air and hydrogen
gas in very different degrees; since it would appear from the
experiments that the ratio of the resistance of hydrogen gas to
that of air is more than double the ratio following from their
densities. Should the existence of such a distinct property of
resistance, varying in the different elastic fluids, be confirmed by
experiments now in progress with other gases, an apparatus more
suitable than the present to investigate the ratio in which it is
possessed by them, could scarcely be devised : and the pendulum,
in addition to its many important and useful purposes in general
physics, may find an application for its very delicate, but, with due
precaution, not more delicate than certain, determinations, in the
domain of chemistry.” Colonel Sabine has informed me that the
experiments here alluded to were interrupted by a cause which
need not now be mentioned, but that as far as they went they
confirmed the result of the experiments with hydrogen, and pointed
out the existence of a specific action in different gases, quite
distinct from mere variations of density.

Our knowledge on the subject of the effect of air on the time
of vibration of pendulums has received a most valuable addition
from the labours of the late Mr Baily, who erected a vacuum
apparatus at his own house, with which he performed many
hundreds of careful experiments on a great variety of pendulums.
The experiments are described in a paper read before the Royal
Society on the 31st of May 1832. The result for each pendulum
is expressed by the value of n, the factor by which the old correc-

1—2

4 ON THE EFFECT OF THE INTERNAL FRICTION OF FLUIDS

tion for buoyancy must be 'multiplied in order to give the whole
effect of the air as deduced from observation. Four spheres,
not quite 1J inch in diameter, gave as a mean n — 1*864, while
three spheres, a little more than 2 inches in diameter, gave only
1*748. The latter were nearly of the same size as those with
which Bessel, by a different method, had obtained & =0-946 or
0*956, which corresponds to ?i = 1*946 or 1*956. Among the
“ Additional Experiments ” in the latter part of Baily’s paper, is a
set in which the pendulums consisted of plain cylindrical rods.
With these pendulums it was found that n regularly increased,
though according to an unknown law, as the diameter of the rod
decreased. While a brass tube 1| inch in diameter gave n equal
to about 2*3, a thin rod or thick wire only 0*072 inch in diameter
gave for n a value as great as 7*530.

Mathematicians in the meanwhile were not idle, and several
memoirs appeared about this time, of which the object was to
determine from hydrodynamics the effect of a fluid on the motion
of a pendulum. The first of these came from the pen of the cele¬
brated Poisson. It was read before the French Academy on the
22nd of August 1831, and is printed in the 11th Volume of the
Memoirs. In this paper, Poisson considers the case of a sphere
suspended by a fine wire, and oscillating in the air, or in any gas.
He employs the ordinary equations of motion of an elastic fluid,
simplified by neglecting the terms which involve the square of the
velocity; but in the end, in adapting his solution to practice, he
neglects, as insensible, the terms by which alone the action of an
elastic differs from that of an incompressible fluid, so that the
result thus simplified is equally applicable to fluids of both classes.
He finds that when insensible quantities are neglected n = 1*5, so
that the mass which we must suppose added to that of the pendu¬
lum is equal to half the mass of the fluid displaced. This result
does not greatly differ from the results obtained experimentally by
Bessel in the case of spheres oscillating in water, but differs ma¬
terially from the result he had obtained for air. It agrees pretty
closely with some experiments which had been performed about
fifty years before by Dubuat, who had in fact anticipated Bessel
in shewing that the time of vibration of a pendulum vibrating in a
fluid would be affected by the inertia of the fluid as well as by its
density. Dubuat’s labours on this subject had been altogether
overlooked by those who were engaged in pendulum experiments;

ON THE MOTION OF PENDULUMS.

5

probably because sucb persons were not likely to seek in a trea¬
tise on hydraulics for information connected with the subject of
their researches. Dubuat had, in fact, rather applied the pen¬
dulum to hydrodynamics than hydrodynamics to the pendulum.

In the Philosophical Magazine for September 1833, p. 185, is a
short paper by Professor Challis, on the subject of the resistance to
a ball pendulum. After referring to a former paper, in which he
had shewn that no sensible error would be committed in a problem
of this nature by neglecting the compressibility of the fluid even if
it be elastic, Professor Challis, adopting a particular hypothesis
respecting the motion, obtains 2 for the value of the factor n for
such a pendulum. This mode of solution, which is adopted in
several subsequent papers, has given rise to a controversy between
Professor Challis and the Astronomer Royal, who maintains the
justice of Poisson's result.

In a paper read before the Royal Society of Edinburgh on the
16th of December 1833, and printed in the 13th Volume of the
Society’s Transactions , Green has determined from the common
equations of fluid motion the resistance to an ellipsoid performing
small oscillations without rotation. The result is expressed by a
definite integral; but when two of the principal axes of the ellip¬
soid become equal, the integral admits of expression in finite
terms, by means of circular or logarithmic functions. When the
ellipsoid becomes a sphere, Green’s result reduces itself to
Poisson’s.

In a memoir read before the Royal Academy of Turin on the
18th of January 1835, and printed in the 37th Volume of the
memoirs of the Academy, M. Plana has entered at great length
into the theory of the resistance of fluids to pendulums. This
memoir contains, however, rather a detailed examination of various
points connected with the theory, than the determination of the
resistance for any new form of pendulum. The author first treats
the case of an incompressible fluid, and then shews that the result
would be sensibly the same in the case of an elastic fluid. In the
case of a ball pendulum, the only one in which a complete solution
of the problem is effected, M. Plana’s result agrees with Poisson’s.

In a paper read before the Cambridge Philosophical Society on
the 29th of May 1843, and printed in the 8th Volume of the
Transactions , p. 105*, I have determined the resistance to a ball
* [Ante, Yol. i. p. 179.]

6 ON THE EFFECT OF THE INTERNAL FRICTION OF FLUIDS

pendulum oscillating within a concentric spherical envelope, and
have pointed out the source of an error into which Poisson had
fallen, in concluding that such an envelope would have no effect.
When the radius of the envelope becomes infinite, the solution
agrees with that which Poisson had obtained for the case of an
unlimited mass of fluid. I have also investigated the increase of
resistance due to the confinement of the fluid by a distant rigid
plane. The same paper contains likewise the calculation of the
resistance to a long cylinder oscillating in a mass of fluid either
unlimited, or confined by a cylindrical envelope, having the same
axis as the cylinder in its position of equilibrium. In the case of
an unconfined mass of fluid, it appeared that the effect of inertia
was the same as if a mass equal to that of the fluid displaced were
distributed along the axis of the cylinder, so that n = 2 in the case
of a pendulum consisting of a long cylindrical rod. This nearly
agrees with Baily’s result for the long 1J inch tube ; but, on com¬
paring it with the results obtained with the cylindrical rods, we
observe the same sort of discrepancy between theory and observa¬
tion as was noticed in the case of spheres. The discrepancy is,
however, far more striking in the present case, as might naturally
have been expected, after what had been observed with spheres, on
account of the far smaller diameter of the solids employed.

A few years ago Professor Thomson communicated to me a
very beautiful and powerful method which he had applied to the
theory of electricity, which depended on the consideration of what
he called electrical images . The same method, I found, applied,
with a certain modification, to some interesting problems relating
to ball pendulums. It enabled me to calculate the resistance to a
sphere oscillating in presence of a fixed sphere, or within a spheri¬
cal envelope, or the resistance to a pair of spheres either in contact,
or connected by a narrow rod, the direction of oscillation being, in
all these cases, that of the line joining the centres of the spheres.
The effect of a rigid plane perpendicular to the direction of motion
is of course included as a particular case. The method even
applies, as Professor Thomson pointed out to me, to the uncouth
solid bounded by the exterior segments of two intersecting spheres,
provided the exterior angle of intersection be a submultiple of two
rig;ht angles. A set of corresponding problems, in which the spheres
are replaced by long cylinders, may be solved in a similar manner.
These results were mentioned at the meeting of the British Asso-

. 'ON THE MOTION OF PENDULUMS.

7

ciation at Oxford in 1847, and are noticed in the volume of reports
for that year, hut they have not yet been published in detail.

The preceding are all the investigations that have fallen under
my notice, of which the object was to calculate from hydrodynamics
the resistance to a body of given form oscillating as a pendulum.
They all proceed on the ordinary equations of the motion of fluids.
They all fail to account for one leading feature of the experimental
results, namely, the increase of the factor w with a decrease in the
dimensions of the body. They recognize no distinction between
the action of different fluids, except what arises from their differ¬
ence of density.

In a conversation with Dr Robinson about seven or eight years
ago on the subject of the application of theory to pendulums, he
noticed the discrepancy which existed between the results of
theory and experiment relating to a ball pendulum, and expressed
to me his conviction that the discrepancy in question arose from
the adoption of the ordinary theory of fluid motion, in which the
pressure is supposed to be equal in all directions. He also de¬
scribed to me a remarkable experiment of Sir James South’s which
he had witnessed. This experiment has not been published, but
Sir James South has kindly allowed me to mention it. "When a
pendulum is in motion, one would naturally have supposed that
the air near the moving body glided past the surface, or the surface
past it, which comes to the same thing if the relative motion only
be considered, with a velocity comparable with the absolute velo¬
city of the surface itself. But on attaching a piece of gold leaf
to the bottom of a pendulum, so as to stick out in a direction per¬
pendicular to the surface, and then setting the pendulum • in
motion, Sir James South found that the gold leaf retained its
perpendicular position just as if the pendulum had been at rest;
and it was not till the gold leaf carried by the pendulum had been
removed to some distance from the surface, that it began to lag
behind. This experiment shews clearly the existence of a tan¬
gential action between the pendulum and the air, and between one
layer of air and another. The existence of a similar action in
water is clearly exhibited in some experiments of Coulomb’s which
will be mentioned in the second part of this paper, and indeed
might be concluded from several very ordinary phenomena. More¬
over Dubuat, in discussing the results of his experiments on the
oscillations of spheres in water, notices a slight increase in the

8 ON THE EFFECT OF THE INTERNAL FRICTION OF FLUIDS

effect of the water corresponding to an increase in the time of
vibration, and expressly attributes it to the viscosity of the fluid.

Having afterwards occupied myself with the theory of the fric¬
tion of fluids, and arrived at general equations of motion, the same
in essential points as those which had been previously obtained in
a totally different manner by others, of which, however, I was not
at the time aware, I was desirous of applying, if possible, these
equations to the calculation of the motion of some kind of pen¬
dulum. The difficulty of the problem is of course materially
increased by the introduction of internal friction, but as I felt
great confidence in the essential parts of the theory, I thought that
labour would not be ill-bestowed on the subject. I first tried a
long cylinder, because the solution of the problem appeared likely
to be simpler than in the case of a sphere. But after having pro¬
ceeded a good way towards the result, I was stopped by a difficulty
relating to the determination of the arbitrary constants, which
appeared as the coefficients of certain infinite series by which the
integral of a. certain differential equation was expressed. Having
failed in the case of a cylinder, I tried a sphere, and presently
found that the corresponding differential equation admitted of
integration in finite terms, so that the solution of the problem
could be completely effected. The result, I found, agreed very
well with Baily’s experiments, when the numerical value of a
certain constant was properly assumed; but the subject was laid
aside for some time. Having afterwards attacked a definite
integral to which Mr Airy had been led in considering the theory
of the illumination in the neighbourhood of a caustic, I found that
the method which I had employed in the case of this ‘integral*
would apply to the problem of the resistance to a cylinder, and it
enabled me to get over the difficulty with which I had before been
baffled. I immediately completed the numerical calculation, so
far as was requisite to compare the formulae with Baily’s experi¬
ments on cylindrical rods, and found a remarkably close agreement
between theory and observation. These results were mentioned at
the meeting of the British Association at Swansea in 1848, and
are briefly described in the volume of reports for that year.

The present paper is chiefly devoted to the solution of the
problem in the two cases of a sphere and of a long cylinder, and to

[Ante, Vol. xi. p. 328.]

ON THE MOTION OP PENDULUMS.

9

a comparison of the results with the experiments of Baily and
others. Expressions are deduced for the effect of a fluid both on
the time and on the arc of vibration of a pendulum consisting
either of a sphere, or of a cylindrical rod, or of a combination of a
sphere and a rod. These expressions contain only one disposable
constant, which has a very simple physical meaning, and which I
propose to call the index of friction of the fluid. This constant we
may conceive determined by one observation, giving the effect of
the fluid either on the time or on the arc of vibration of any one
pendulum of one of the above forms, and then the theory ought to
predict the effect both on the time and on the arc of vibration of
all such pendulums. The agreement of theory with the experi¬
ments of Baily on the time of vibration is remarkably close. Even
the rate of decrease of the arc of vibration, which it formed no part
of Baily’s object to observe, except so far as was necessary for
making the small correction for reduction to indefinitely small
vibrations, agrees with the result calculated from theory as nearly
as could reasonably be expected under the circumstances.

It follows, from theory that with a given sphere or cylindrical
rod the factor n increases with the time of vibration. This accounts
in a good measure for the circumstance that Bessel obtained so
large a value of k for air, as is shewn at length in the present
paper; though it unquestionably arose in a great degree from the
increase of resistance due to the close proximity of a rigid plane to
the swinging ball.

I have deduced the value of the index of friction of water from
some experiments of Coulomb’s on the decrement of the arc of
oscillation of disks, oscillating in water in their own plane by the
torsion of a wire. When the numerical value thus obtained is
substituted in the expression for the time of vibration of a sphere,
the result agrees almost exactly with Bessel’s experiments with a
sphere swung in water.

The present paper contains one or two applications of the
theory of internal friction to problems which are of some interest,
but which do not relate to pendulums. The resistance to a sphere
moving uniformly in a fluid may be obtained as a limiting case of
the resistance to a ball pendulum, provided the circumstances be
such that the square of the velocity may be neglected. The resist¬
ance thus determined proves to be proportional, for a given fluid
and a given velocity, not to the surface, but to the radius of the

10 ON THE EFFECT OF THE INTERNAL FRICTION OF FLUIDS

sphere; and therefore the accelerating force of the resistance in¬
creases much more rapidly, as the radius of the sphere decreases,
than if the resistance varied as the surface, as would follow from
the common theory. Accordingly, the resistance to a minute
globule of water falling through the air with its terminal velocity
depends almost wholly on the internal friction of air. Since the
index of friction of air is known from pendulum experiments,
we may easily calculate the terminal velocity of a globule of given
size, neglecting the part of the resistance which depends upon the
square of the velocity. The terminal velocity thus obtained is so
small in the case of small globules such as those of which we may
conceive a cloud to be composed, that the apparent suspension of
the clouds does not seem to present any difficulty. Had the re¬
sistance been determined from the common theory, it would have
been necessary to suppose the globules much more minute, in
order to account in this way for the phenomenon. Since in the
case of minute globules falling with their terminal velocity the
part of the resistance depending upon the square of the velocity, as
determined by the common theory, is quite insignificant compared
with the part which depends on the internal friction of the air, it
follows that were the pressure equal in all directions in air in the
state of motion, the quantity of water which would remain sus¬
pended in the state of cloud would be enormously diminished. The
pendulum thus, in addition to its other uses, affords us some inter¬
esting information relating to the department of meteorology.

The fifth section of the first part of the present paper contains
an investigation of the effect of the internal friction of water in
causing a series of oscillatory waves to subside. It appears from
the result that in the case of the long swells of the ocean the effect
of friction is insignificant, while in the case of the ripples raised by
the wind on a small pool, the motion subsides very r’apidly when
the disturbing force ceases to act.

ON THE MOTION OF PENDULUMS.

11

PART I.

ANALYTICAL INVESTIGATION.

Section I.

Adaptation of the general equations to the case of the fluid sur¬
rounding a body which oscillates as a pendulum . General
laws which follow from the form of the equations. Solution
of the equations in the case of an oscillating plane.

1. In a paper “ On the Theories of the Internal Friction of
Fluids in Motion , &c which the society did me the honour to
publish in the 8th Volume of their Transactions, I have arrived
at the following equations for calculating the motion of a fluid
when the internal friction of the fluid itself is taken into account,
and consequently the pressure not supposed equal in all directions :

dp

dx

du

dt

du
' dx

du du
' v dy~ w dz.

+ f*

/ d 2 u d 2 u
\dx 2 dy 2

+

d 2 u\

df)

, n A

3 dx

/du dv
yfec + dy

dw'
dz ,

( 1 ),

with two more equations which may be written down from
symmetry. In these equations u , v, w are the components of the
velocity along the rectangular axes of x, y } z\ X , YZ are the
components of the accelerating force; p is the pressure, t the time,
p the density, and p a certain constant depending on the nature
of the fluid.

The three equations of which (1) is the type are not the
general equations of motion which apply to a heterogeneous fluid
when internal friction is taken into account, which are those
numbered 10 in my former paper, but are applicable to a homo¬
geneous incompressible fluid, or to a homogeneous elastic fluid
subject to small variations of density, such as those which accom¬
pany sonorous vibrations. It must be understood to be included in
the term homogeneous that the temperature is uniform throughout

[ Ante , Yol. i. p. 75.]

12 OK THE EFFECT OF THE INTERNAL FRICTION OF FLUIDS

the mass, except so far as it may be raised or lowered by sudden
condensation or rarefaction in the case of an elastic fluid. The
general equations contain the differential coefficients of the quan¬
tity fjb with respect to x , y, and but the equations of the form (1)
are in their present shape even more general than is required for
the purposes of the present paper.

These equations agree in the main with those which had been
previously obtained, on different principles, by Navier, by Poisson,
and by M. de Saint-Yenant, as I have elsewhere observed*. The
differences depend only on the coefficient of the last term, and
this term vanishes in the case of an incompressible fluid, to which
Navier had confined his investigations.

e qnat,io ns such as (1) in their present shape are rather„.
complicated, but in applying them to the case of a pendulum they
may be a good deal simplified without the neglect of any quan¬
tities which it would be important to retain. In the first place
the motion is supposed very small, on which account it will be
allowable to neglect the terms which involve the square of the
velocity. In the second place, the nature of the motion that we
have got to deal with is such that the compressibility of the fluid
has very little influence on the result, so that we may treat the
fluid as incompressible, and consequently omit the last terms in
the equations. Lastly, the forces X, Y, Z are in the present case
the components of the force of gravity, and if we write

p + TL + pf(Xdx+ Ydy + Zdz)
for p> we may omit the terms X, Y, Z.

If / be measured vertically downwards from a horizontal
plane drawn in the neighbourhood of the pendulum, and if g be
the force of gravity, f(Xdx + Ydy -f Zdz) = gz\ the arbitrary
constant, or arbitrary function of the time if it should be found
necessary to suppose it to be such, being included in IT. The
part of the w T hole force acting on the pendulum which depends on
the terms H-\-gpz is simply a force equal to the weight of the
fluid displaced, and acting vertically upwards through the centre
of gravity of the volume.

* Report on recent researches in Hydrodynamics. Report of the British Asso¬
ciation for 1846, p. 16. [Ante, Yol. i. p. 182.]

ON THE MOTION OF PENDULUMS.

13

When simplified in the manner just explained, the equations
such as (1) become

dp __ / d 2 u d 2 u d 2 u \ du ~

dx ^ \ dx l dy 2 dz 2 ) ^ dt

dp __ fd 2 v d 2 v d 2 v\ __ dv

dy ^ \dx 2 dy 2 dz 2 J ^ dt

dp __ /d 2 w d 2 w d 2 w \ __ dw

dz ^ \dx 2 dy 2 + dz 2 J ^ dt ^

which, with the equation of continuity, ‘

( 2 ),

•( 3 ),

du ^dv ^dw
dx + dy^ dz~

are the only equations which have to be satisfied at all points of
the fluid, and at all instants of time.

In applying equations (2) to a particular pendulum experiment,
we may suppose y constant; but in order to compare experiments
made in summer with experiments made in winter, or experiments
made under a high barometer with experiments made under a
low, it will be requisite to regard p as a quantity which may vary
with the temperature and pressure of the fluid. As far as the
result of a single experiment*, which has been already mentioned,
performed with a single elastic fluid, namely air, justifies us in
drawing such a general conclusion, we may assert that for a given
fluid at a given temperature y, varies as pf.

2. For the formation of the equations such as (1), I must
refer to my former paper; but it will be possible, in a few words,
to enable the reader to form a clear idea of the meaning of the
constant y.

Conceive the fluid to move in planes parallel to the plane of
xy , the motion taking place in a direction parallel to the axis of y.
The motion will evidently consist of a sort of continuous sliding,
and the differential coefficient dvjdz may be taken as a measure of

* The first of the experiments described in Col. Sabine’s paper, in which the
gauge stood as high as 7 inches, leads to the same conclusion; but as the vacuum
apparatus had not yet been made stanch it is perhaps hardly safe to trust this
experiment in a question of such delicacy.

+ [We now know that ix is independent of p, until excessive exhaustions are
reached, far beyond any that we have here to deal with.]

14 ON THE EFFECT OF THE INTERNAL FRICTION OF FLUIDS

the rate of sliding. In the theory it is supposed that in general
the pressure about a given point is compounded of a normal
pressure, corresponding to the density, which being normal is
necessarily equal in all directions, and of an oblique pressure or
tension, altering from one direction to another, which is expressed
by means of linear functions of the nine differential coefficients of
the first order of u> v , w with respect to x , y , z , which define the
state of relative motion at any point of the fluid. Now in the
special case considered above, if we confine our attention to one
direction, that of the plane of xy, the total pressure referred to a
unit of surface is compounded of a normal pressure corresponding
to the. density, and a tangential pressure expressed by pdv/dz ,
which tends to reduce the relative motion.

In the solution of equations (2), p always appears divided
by p . Let p~p!(>. The constant p may conveniently be called
the indeoc of friction of the fluid, whether liquid or gas, to which
it relates. As regards its dimensions, it expresses a moving force
divided by the product of a surface, a density, and the differential
coefficient of a velocity with respect to a line. Hence p is the
square of a line divided by a time, whence it will be easy to adapt
the numerical value of p' to a new unit of length or of time.

3. Besides the general equations (2) and (3), it will be
requisite to consider the equations of condition at the boundaries
of the fluid. For the purposes of the present paper there will be
no occasion to consider the case of a free surface, but only that of
the common surface of the fluid and a solid. Now, if the fluid
immediately in contact with a solid could flow past it with a
finite, velocity, it would follow that the solid was infinitely smoother
with respect to its action on the fluid than the fluid with
respect to its action on itself. For, conceive the elementary
layer of fluid comprised between the surface of the solid and a
parallel surface at a distance h , and then regard only so much of
this layer as corresponds to an elementary portion dS of the
surface of the solid. The impressed forces acting on the fluid
element must be in equilibrium with the effective forces reversed.
Now conceive h to vanish compared with the linear dimensions
of dS, and lastly let dS vanish*. It is evident that the conditions

. * To be quite precise it would be necessary to say, Conceive h and dS to vanish

together, h vanishing compared with the linear dimensions of dS ; for so long as dS

ON THE MOTION OF PENDULUMS,

15

of equilibrium will ultimately reduce themselves to this, that
the oblique pressure which the fluid element experiences on
the side of the solid must be equal and opposite to the pressure
which it experiences on the side of the fluid. Now if the fluid
could flow past the solid with a finite velocity, it would follow
that the tangential pressure called into play by the continuous
sliding of the fluid over itself was no more than counteracted
by the abrupt sliding of the fluid over the solid. As this appears
exceedingly improbable a priori , it seems reasonable in the
first instance to examine the consequences of supposing that
no such abrupt sliding takes place, more especially as the ma¬
thematical difficulties of the problem will thus be materially
diminished. I shall assume, therefore, as the conditions to be
satisfied at the boundaries of the fluid, that the velocity of a fluid
particle shall be the same, both in magnitude and direction, as
that of the solid particle with which it is in contact. The agree¬
ment of the results thus obtained with observation will presently
appear to be highly satisfactory. When the fluid, instead of being
confined within a rigid envelope, extends indefinitely around the
oscillating body, we must introduce into the solution the condition
that the motion shall vanish at an infinite distance, which takes
the place of the condition to be satisfied at the surface of the
envelope.

To complete the determination of the arbitrary functions which
would be contained in the integrals of (2) and (3), it would be
requisite to put t = 0 in the general expressions for u, v, w, obtained
by integrating those equations, and equate the results to the initial
velocities supposed to be given. But it would be introducing a
most needless degree of complexity into the solution to take
account of the initial circumstances, nor is it at all necessary to do
so for the sake of comparison of theory with experiment. For in
a pendulum experiment the pendulum is set swinging and then
left to itself, and the first observation is not taken till several
oscillations have been completed, during which any irregularities
attending the initial motion would haye had time to subside. It

remains finite we cannot suppose h to vanish altogether, on account of the curva¬
ture of the elementary surface. Such extreme precision in unimportant matters
tends, I think, only to perplex the reader, and prevent him from entering so readily
into the spirit of .an investigation.

16 ON THE EFFECT OF THE INTERNAL FRICTION OF FLUIDS

will be quite sufficient to regard the motion as already going on,
and limit the calculation to the determination of the simultaneous
periodic movements of the pendulum and the surrounding fluid.
The arc of oscillation will go on slowly decreasing, but it will be
so nearly constant for several successive oscillations that it may be
regarded as strictly such in calculating the motion of the fluid;
and having thus determined the resultant action of the fluid on
the solid we may employ the result in calculating the decrement
of the arc of oscillation, as well as in calculating the time of oscil¬
lation. Thus the assumption of periodic functions of the time in
the expressions for u } v, w will take the place of the determination
of certain arbitrary functions by means of the initial circumstances.

4. Imagine a plane drawn perpendicular to the axis of x
through the point in the fluid whose co-ordinates are x, y , z. Let
the oblique pressure in the direction of this plane be decomposed
into three pressures, a normal pressure, which will be in the
direction of x, and two tangential pressures in the directions of y, z,
respectively. Let P t be the normal pressure, and T z the tangential
pressure in the direction of y, which will be equal to the component
in the direction of x of the oblique pressure on a plane drawn per¬
pendicular to the axis of y. Then by the formulae (7), (8) of my
former paper, and (3) of the present,

- p .=f- 2 4.“>•

rp __ (& u , &V'

8 ^ \dy dx ,

(5).

These formulae will be required in finding the resultant force of
the fluid on the pendulum, after the motion of the fluid has been
determined in terms of the quantities by which the motion of the
pendulum is expressed.

5. Before proceeding to the solution of the equations (2) and
(3) in particular cases, it will be well to examine the general laws
which follow merely from the dimensions of the several terms
which appear in the equations.

Consider any number of similar systems, composed of similar
solids oscillating in a similar manner in different fluids or in the
same fluid. Let a , a\ a"... be homologous lines in the different

ON THE MOTION OF PENDULUMS.

17

systems ; T : T',! 7 "... corresponding times, such for example as the
times of oscillation from rest to rest. Let x, y, z be measured from
similarly situated origins, and in corresponding directions, and t
from corresponding epochs, such for example as the commencements
of oscillations when the systems are beginning to move from a
given side of the mean position.

The form of equations (2), (3) shews that the equations being
satisfied for one system will be satisfied for all the systems provided

i LbU DUX

u oc v oc w 3 x oc y oc z, and j)oc — oc .

X v

The variations x oc y oc z merely signify that we must compare
similarly situated points in inferring from the circumstance that
(2), (3) are satisfied for one system that they will be satisfied for
all the systems. If c, c', c"... be the maximum excursions of simi¬
larly situated points of the fluids

c

WOCy,

x cc a, tcc T>

and the sole condition to be satisfied, in addition to that of geo¬
metrical similarity, in order that the systems should be dynamically
similar, becomes

d 2 a ,

m OC - Or OC /£,.(6).

r

This condition being satisfied, similar motions will take place in
the different systems, and we shall have

pac

( 7 ).

It follows from the equations (4), (5), and the other equations
which might be written down from symmetry, that the pressures
such as P v T 3 vary in the same manner as p, whence it appears
from (7) that the resultant or resultants of the pressures of the
fluids on the solids, acting along similarly situated lines, which
vary asjpa 2 , vary as pa 3 and cT ~ 2 conjointly. In other words, these
resultants in two similar systems are to one another in a ratio com¬
pounded of the ratio of the masses of fluid displaced, and of the
ratio of the maximum accelerating effective forces belonging to
similarly situated points in the solids.

6. In order that two systems should be similar in which the
fluids are confined by envelopes that are sufficiently narrow to

s. m.

3 ON THE EFFECT OF THE INTERNAL FRICTION OF FLUIDS

mfluence the motion of the fluids, it is necessary that the envelopes
should be similar and similarly situated with respect to the solids
oscillating within them, and that their linear dimensions should
be in the same ratio as those of the oscillating bodies. In strict¬
ness, it is likewise necessary that the solids should be similarly
situated with respect to the axis of rotation. If however two
similar solids, such as two spheres, are attached to two fine
wires, and made to perform small oscillations in two unlimited
masses of fluid, and if we agree to neglect the effect of the sus¬
pending wires, and likewise the effect of the rotation of the
spheres on the motion of the fluid, which last will in truth be
exceedingly small, we may regard the two systems as geometri¬
cally similar, and they will be dynamically similar provided the
condition (6) be satisfied. When the two fluids are of the same
nature, as for example when both spheres oscillate in air, the
condition of dynamical similarity reduces itself to this, that the
times of oscillation shall be as the squares of the diameters of the
spheres.

If, with Bessel, we represent the effect of the inertia of the
fluid on the time of oscillation of the sphere by supposing a mass
equal to 1c times that of the fluid displaced added to the mass of
the sphere, which increases its inertia without increasing its weight,
we must expect to find Tc dependent on the nature of the fluid,
and likewise on the diameter of the sphere. Bessel, in fact, ob¬
tained very different values of k for water and for air. Baily’s
experiments on spheres of different diameters, oscillating once in
a second nearly, shew that the value of k increases when the
diameter of the sphere decreases. Taking this for the present as
the result of experiment, we are led from theory to assert that
the value of k increases with the time of oscillation; in fact, Jc
ought to be as much increased as if we had left the time of oscil¬
lation unchanged, and diminished the diameter in the ratio in
which the square root of the time is increased. It may readily
be shewn that the value of k obtained by Bessel’s method, k by
means of a long and short pendulum, is greater than what belongs
to the long pendulum, much more, greater than what belongs to
the shorter pendulum, which oscillated once in a second nearly.
The value of k given by Bessel is in fact considerably larger than
that obtained by Baily, by a direct method, from a sphere of nearly

ON THE MOTION OF PENDULUMS.

19

the same size as those employed by Bessel, oscillating once in a
second nearly.

The discussion of the experiments of Baily and Bessel belongs
to Part II. of this paper. They are merely briefly noticed here to
shew that some results of considerable importance follow readily
from the general equations, even without obtaining any solution
of them.

7. Before proceeding to the problems which mainly occupy
this paper, it may be well to exhibit the solution of equations (2)
and (3) in the extremely simple case of an oscillating plane.

Conceive a physical plane, which is regarded as infinite, to be
situated in an unlimited mass of fluid, and to be performing small
oscillations in the direction of a fixed line in the plane. Let a
fixed plane coinciding with the moving plane be taken for the
plane of yz i the axis of y being parallel to the direction of motion,
and consider only the portion of fluid which lies on the positive
side of the plane of yz. In the present case, we must evidently
have u = 0, w = 0; and p, v will be functions of x and t , which
have to be determined. The equation (3) is satisfied identically,
and we get from (2), putting /*=/*'/},

dp

dx

= 0 ,

dv _ , d 2 v
dt ^ dx z

( 8 ).

The first of these equations gives p = a constant, for it evidently
cannot be a function of £, since the effect of the motion vanishes
at an infinite distance from the plane; and if we include this
constant in II, we shall havejp = 0. Let Fbe the velocity of the
plane itself, and suppose

F=csinw£.(9).

Putting in the second of equations (8)

v = X x sin nt + X 2 cos nt
, d 2 X a v ,<PX l

we get

nX t =

n dx 4

( 10 ),

( 11 ).

The last of these equations gives

= e -J£t x (Asm + £ cos J^x)

+ (C sin J 2 ^ + -° cos J^j' x ) ■

2—2

20 ON THE EFFECT OF THE INTERNAL FRICTION OF FLUIDS

Since X 2 must not become infinite when x = oo, we must have
0=0, D = 0. Obtaining X L from the first of equations (11), and
substituting in (10), we get

® = e - ^'* j— A sin (nt — \J cos '

V

Now by the equations of conditions assumed in Art. 3, we must
have v=V when x- 0, whence

n

2fjb'

To find the normal and tangential components of the pressure
of the fluid on the plane, we must substitute the above value of v
in the formulae (4), (5), and after differentiation put x — 0. P i;
T % will then be the components of the pressure of the solid on the
fluid, and therefore -P v - T z , those of the pressure of the fluid
on the solid. We get

P x =0, T = c P/ ^/^- (sinnt+cosnt)=py / ^ (V+i^Q...(13).

-,se „

D = ce v 2 ix f sm

■( nt “V

The force expressed by the first of these terms tends to diminish
the amplitude of the oscillations of the plane. The force expressed
by the second has the same effect as increasing the inertia of the
plane.

8. The equation (12) shews that a given phase of vibration is
propagated from the plane into the fluid with a velocity
while the amplitude of oscillation decreases in geometric progres¬
sion as the distance from the plane increases in arithmetic. If
we suppose the time of oscillation from rest to rest to be one
second, n — ir\ and if we suppose yV“’ 116 inch, which, as will
presently be seen, is about its value in the case of air, we get for
the velocity of propagation *2908 inch per second nearly. If we
enquire the distance from the plane at which the amplitude of
oscillation is reduced to one'half, we have only to put

V(w/2/Z) a? = log,2,

which gives, on the same suppositions as before respecting nu¬
merical values, x— ‘06415 inch nearly. For water the value of \j!
is a good deal smaller than for air, and the corresponding value of a;
smaller likewise, since it varies cceteris paribus as y^- Hence if

ON THE MOTION OF PENDULUMS.

21

a solid of revolution of large, or even moderately large, dimensions
be suspended by a fine wire coinciding with the axis of revolution,
and made to oscillate by the torsion of the wire, the effect of the
fluid may be calculated with a very close degree of approximation
by regarding each element of the surface of the solid as an element
of an infinite plane oscillating with the same linear velocity*.

For example, let a circular disk of radius a be suspended
horizontally by a fine wire attached to the centre, and made to
oscillate. Let r be the radius vector of any element of the disk,
measured from its centre, 0 the angle through which the disk has
turned from its mean position. Then in equation (13), we must
put V = r ddjdt , whence

dO ldM\

dt n dt 2 )

The area of the annulus of the disk comprised between the radii r
and r + dr is 4tfrrdr } both faces being taken, and if Q be the whole

moment of the force of the fluid on the disk, G = — 4nr L r 2 T s dr ,

j 0

whence

G~ — irpcf

d6 1 d 2 d\
dt^~ n df)

Let My 1 be the moment of inertia of the disk, and let n x be what n
would become if the fluid were removed, so that — n 2 Mrf6 is the
moment of the force of torsion. Then when the fluid is present
the equation of motion of the disk becomes

+ + g + <Jf^ = 0...(14),

or, putting for shortness

f+<«=o.

which gives, neglecting /3 2 ,

6= 0 o €~ n ^sia (nt+ a) .(15),

where n = oi 1 (1 — /3).

* [That is, of course, on the supposition that the oscillations arc not excessively
slow.]

22 ON THE EFFECT OF THE INTERNAL FRICTION OF FLUIDS

The observation of n and n lf or else the observation of n and
of the decrement of the arc of oscillation, would enable us to de¬
termine ft, and tbence p. The values of ft determined in these
two different ways ought to agree.

There would be no difficulty in obtaining a more exact solution,
in which the decrement of the arc of oscillation should be taken
into account in calculating the motion of the fluid, but I pass on
to the problems, the solution of which forms the main object of
this paper.

Section II

Solution of the equations in the case of a sphere oscillating in a
mass of fluid either unlimited } or confined by a spherical
envelope concentric with the sphere in its position of equi¬
librium.

9. Suppose the sphere suspended by a fine wire, the length
of which is much greater than the radius of the sphere. Neglect
for the present the action of the wire on the fluid, and consider
only that of the sphere. The motion of the sphere and wire being
supposed to take place parallel to a fixed vertical plane, there are
two different modes of oscillation possible. We have here nothing
to do with the rapid oscillations which depend mainly on the
rotatory inertia of the sphere, but only with the principal oscil¬
lations, which are those which are observed in pendulum ex¬
periments. In these principal oscillations the centre of the sphere
describes a small arc of a curve which is very nearly a circle, and
which would be rigorously such, if the line joining the centre of
gravity of the sphere and the point of attachment of the wire
were rigorously in the direction of the wire. In calculating the
motion of the fluid, w r e may regard this arc as a right line. In
fact, the error thus introduced would only be a small quantity of
the second order, and such quantities are supposed to be neglected
in the investigation. Besides its motion of translation, the sphere
will have a motion of rotation about a horizontal axis, the angular
motion of the sphere being very nearly the same as that of the
suspending wire. This motion, which would produce absolutely
no effect on the fluid according to the common theory of hydro-

ON THE MOTION OF PENDULUMS.

23

dynamics, will not be without its influence when friction is taken
into account; but the effect is so very small in practical cases that
it is not worth while to take it into account. For if a be the
radius of the sphere, and l the length of the suspending wire, the
velocity of a point in the surface of the sphere due to the motion
of rotation will be a small quantity of the order a/l compared with
the velocity due to the motion of translation. In finding the
moment of the pressures of the fluid on the pendulum, forces
arising from these velocities, and comparable with them, have
to be multiplied by lines which are comparable with a, l , respec¬
tively. Hence the moment of the pressures due to the motion of
rotation of the sphere will be a small quantity of the order a 2 /P,
compared with the moment due to the motion of translation.
Now in practice l is usually at least twenty or thirty times greater
than a , and the whole effect to be investigated is very small,
so that it would be quite useless to take account of the motion
of rotation of the sphere.

The problem, then, reduces itself to this. The centre of a
sphere performs small periodic oscillations along a right line, the,
sphere itself having a motion of translation simply: it is required
to determine the motion of the surrounding fluid.

10. Let the mean position of the centre of the sphere be taken
for origin, and the direction of its motion for the axis of x, so that
the motion of the fluid is symmetrical with respect to this axis.
Let vr be the perpendicular let fall from any point on the axis
of x } q the velocity in the direction of <nr, co the angle between the
line ‘sr and the plane of xij. Then p } u, and q will be functions
of 'zzr, and t, and we shall have

v = q cos co, w = q sm co, y = m cos co, z = w sin co,

whence

«r 2 = y* 4- z 2 , co = tan 1 - .

We have now to substitute in equations (2) and (3), and we
are at liberty to put co = 0 after differentiation. We get

d d sm co d d

~Y~ 2=5 cos co —j - r _ = when co = 0.

(iy dur w dco dur

d 2 d 2 ,

d? = d? yrheno = 0 ’

14 ON THE EFFECT OF THE INTERNAL FRICTION OF FLUIDS

d . d cos w d Id, A

- r = sinw- r -)- r- , = — j- when ® = U,

dz dtss 13 aw 13 aw

d 2 1 d 1 d 2 ,
whence we obtain

dp (d 2 u d 2 u 1 du\ du

dx ^ \dx 2 dm 2 m dvr) ^ dt

_ „ (^1 . .1 II _ JL\_ n f 17^

dm ^ \dx 2 dm 2 m dm -sr 2 ; ? dt .

^ + ^£ + £ = 0.(18).

dx dzr m

Eliminating p from (16) and (17), and putting for jjl its equi¬
valent fip } we get

M di3 W + di3* + 13 dm) U P dx W + dtj 2

1 d 1 \ __ d (du __ dq\ _ ~

Z T dzT 'ST / ^ dt \Cfcr dx)

( d^_ cV^ , 1 d __ 1 1 d \ (d/w _ dq\ _ 0 ^

01 \dx 2 dm 2 m dm 'st 2 p dt) \dm dx)

By virtue of (18), m (udz r — qdx) is an exact differential. Let
then

m {udm — qdx) = d^r .(20).

Expressing u and q in terms of we get

__ <2^ _ 1 / c? 2 d * I, 'dr

dm dx m \doc 2 dm 2 m dm)

Substituting in (19), and operating separately on the factor m~\
we obtain

l^L i j^L __ 1 A __ _L ^_L ^ A A\ f -n f9rn

V& 2 cfe 2 i3T 6?sr p! dt) \da? dm 2 -stcZst/^

Since the operations represented by the two expressions within
parentheses are evidently convertible, the integral of this equation is

t = + .(21),

* If we denote for shortness the operation

dP_ JP_ _1J L
dx 2 ^ dm 2 m dm

ON THE.MOTION OF PENDULUMS.

25

•where fa, fa are the integrals of the equations

fd* , 1 n

V<fe 2 <fe 2 w cfov ' 1

1 1 (?\ i A

(&_ d*_

\dx 2 + dzr 2 '

( 22 ),

(23).

11. By means of the last three equations, the expression for
dp obtained from (16) and (17) is greatly simplified. We get, in
the first place,

1 - f / (#_ . , 1 _ ^.1 r24V

p cfo \c?# 2 cfc 2 13J dus) dt J 'us d'us .^

but by adding together equations (22) and (23), and taking
account of (21), -we get

d 2 ^\r __ cZ 2 ^ _1 (hjf 1^ dfa

dx 2 dus 2 us dus /jl dt

On substituting in (24), it will be found that all the terms in the
right-hand member of the equation destroy one another, except
those which contain dtyjdt and dfa/dt, and the equation is re¬
duced to

dp __ p d 2 fa

dx 'us dtd'us *

by D, our equation becomes

which gives by the separation of symbols

- JD -1 | 0

H

so that dx/z/dt is composed of two parts, which are separately the integrals of (22),
(23). Hence we have for the integral of (20') ^=^ + ^0 + ^, SP" being a function of
x and 'us without t which satisfies the equation D 2 \I> = 0. For the equations (22),
(23) will not be altered if we put ffadt, ffadt for \p. 2 , the arbitrary functions
which would arise from the integration with respect to t being supposed to be in¬
cluded in Sk. The function Sk, which taken by itself can only correspond to steady
motion, is excluded from the problem under consideration by the condition of
periodicity. But we may even, independently of this condition, regard (21) as the
complete integral of (20'), provided we suppose included in (21) terms which would
be obtained by supposing \Jz at first to vary slowly with the time, employing the
integrals of (22) and (23), and then making the rate of variation diminish inde¬
finitely. By treating the symbolical expression in the right-hand member of equa¬
tion (a) as a vanishing fraction, djdt being supposed to vanish, we obtain in fact
D~ 2 0; so that under the convention just mentioned the function SI> may be sup¬
posed to be included in The same remarks will apply to the equation in

Section III. which answers to (20').

26 ON THE EFFECT OF THE INTERNAL FRICTION OF FLUIDS

The equation (17) may be reduced in a similar manner, and
we get finally

£*). (25)>

which is an exact differential by virtue of (22).

12. Passing to polar co-ordinates, let r be the radius vector
drawn from the origin, 0 the angle which r makes with the axis of
x , and let R be the velocity along the radius vector, © the velocity
perpendicular to the radius vector : then

x = r cos 6, nr — r sin 0 } u = B cos 0 — ® sin 0 } q = B sin 0 H- © cos 0.
Making these substitutions in (20), (22), (23), and (25), we obtain
r sin 0 ( Brd0 — ®dr) = dty .(26),

sin 0 d / 1 ___ 0

dr 2 r 2 dd Isin 6 d0 )

(27),

+ sin<? §l ( 1 1 ^ ! = Q

dr*

r* dd \sin 6 d0 )

fi dt

dv - R - ( r d0 — - ^ ^ dr
d P~~r sin 0 \dtdr r dtd0 d

(28),

(29).

We must now determine ^ and by means of (27) and (28),
combined with the equations of condition. When these functions
are known, p will be obtained by integrating the exact differential
which forms the right-hand member of (29), and the velocities
R, ©, if required, will be got by differentiation, as indicated by
equation (26). Pormulse deduced from (4) and (5) will then make
known the pressure of the fluid on the sphere.

13. Let £ be the abscissa of the centre of the sphere at any
instant. The conditions to be satisfied at the surface of the sphere
are that when r = r v the radius vector of the surface, we have

-8 = cosd^|, ©=-sin 8^..
dt dt

Now r 1 differs from a by a small quantity of the first order, and
since this value of r has to be substituted in functions which are
already small quantities of that order, it will be sufficient to put
r = a. Hence, expressing R and © in terms of -fr, we get

= a sin 2 8 ^^ = a 2 sin 8 cosd whenr=a.(30).

ON THE MOTION OF PENDULUMS.

27

When the fluid is unlimited, it will he found that certain
arbitrary constants will vanish by the condition that the motion
shall not become infinite at an infinite distance in the fluid. When
the fluid is confined by an envelope having a radius b , we have the
equations of condition

= 0, = 0, when r=b .(31).

dr ad

14. We must now, in accordance with the plan proposed in
Section I., introduce the condition that the function ^ shall be
composed, so far as the time is concerned, of the circular functions
sin nt and cos nt, that is, that it shall be of the form

P sin nt 4* Q cos nt,

where P and Q are functions of r and 9 only. An artifice, how¬
ever, which has been extensively employed by M. Cauchy will
here be found of great use. Instead of introducing the circular
functions sin nt and cos nt, we may employ the exponentials
and g-V-ini Since our equations are linear, and since each of
these exponential functions, reproduces itself at each differentia¬
tion, it follows that if all the terms in any one of our equations be
arranged in two groups, containing as a factor e^“ ln * in one case,
and e~^~ lnt in the other, the two groups will be quite independent,
and the equations will be satisfied by either group separately.
Hence it will be sufficient to introduce one of the exponential
functions. We shall thus have only half the number of terms to
write down, and half the number of arbitrary constants to deter¬
mine that would have been necessary had we employed circular
functions. When we have arrived at our result, it will be sufficient
to put each equation under the form JJ + J — 1 F= 0 , and throw
away the imaginary part, or else throw away the real part and
omits/—1, since the system of quantities U, and the system of
quantities V must separately satisfy the equations of the problem.
Assuming then

nt

^ = e' J ~ lnt P,

we have to determine P as a function of r and 9 .

28 ON THE EFFECT OF THE INTERNAL FRICTION OF FLUIDS

15. The form of the equations of condition (30) points out

sin 2 # as a factor of P, and since the operation sin # ^ ^

performed on the function sin 2 # reproduces the same function with
a coefficient — 2, it will be possible to satisfy equations (27) and
(28) on the supposition that sin 2 # is a factor of fa and fa*-
Assume then

fa = e^~ lnt sin 2 9f 1 (r), fa = sin 2 0f 2 (r).

Putting for convenience

n J — 1 = fim? .(32),

and substituting in (27) and (28), we get

/ 1 »- 2 ,/ 1 (0 = 0 .(33),

/.»--*»*/,(»•) = 0 .(34).

The equations of condition (30), (31) become, on putting/(r)
for /,(?*) +/ t (r),

/' (a) = ac, f{a) = \a?c .(35),

/'(&) = 0, /(&) = 0 .(36).

We may obtain p from (29) by putting for fa its value
e y.’m?t s replacing after differentiation 2!/j(r) by its equiva-

* When this operation is performed on the function sin0 dYJdd, the function is
reproduced with a coefficient — i (i + 1). F* here denotes a Laplace’s function of the
I th - order, which contains only one variable angle, namely 0. Now p may be ex¬
panded in a series of quantities of the general form si nddYJdd. For, since we are
only concerned with the differential coefficients of p with respect to r and 0, we have
a right to suppose ip to vanish at whatever point of space we please. Let then
ip—0 when r—a and 0 = 0. To find the value of \p at a distance r from the origin,
along the axis of x positive, it will be sufficient to put 0 = 0, <20 = 0 in (20), and
integrate from r = a to r, whence \p=.0. To find the value of ip at the same dis¬
tance r along the axis of x negative, it will he sufficient to leave r constant, and
integrate dip from 0 = 0 to 0 = 7r. Referring to (26), we see that the integral vanishes,
since the total flux across the surface of the sphere whose radius is r must be equal
to zero. Hence ip vanishes when 0 = 0 or =tt, and it appears from (26) that when
8 is very small or very nearly equal to x, ip varies ultimately as sin 2 0 for given
values of r and f. Hence ip cosec 0, and therefore fp cosec 0 dd, is finite even when
sin0 vanishes, and therefore/^ cosec0 dd may be expanded in a series of Laplace’s
functions, and therefore p itself in a series of quantities of the form sin0 dYi/dd.
It was somewhat in this way that I first obtained the form of the function p.

ON THE MOTION OF PENDULUMS.

29

lent 'r*f”(r)> ai3 d ^ en integrating. It is unnecessary to add an
arbitrary function of the time, since any sucb function may be
supposed to be included in II. We get

p = — cos 0// (r) .(37).

16. The integration of the differential equation (33) does not
present the least difficulty, and (34) comes under a well-known
integrable form. The integrals of these equations are

• /n n \

Ur) = Ce™ (1 + — J + De mr (1 - — J j

and we have to determine A } B } G ' D by the equations of con¬
dition.

The solution of the problem, in the case in which the fluid is
confined by a spherical envelope, will of course contain as a par¬
ticular case that in which the fluid is unlimited, to obtain the
results belonging to which it will be sufficient to put b = oo . As,
however, the case of an unlimited fluid is at the same time simpler
and more interesting than the general case, it will be proper to
consider it separately.

Let + m denote that square root of pu~ x n J — 1 which has its
real part positive; then in equations (38) we must have D = 0,
since otherwise the velocity would be infinite at an infinite dis¬
tance. We must also have 5 = 0, since otherwise the velocity
would be finite when r = oo, as appears from (26). We get then
from the equations of condition (35)

A 1 3 3 a?c A , 1 \ n Sac ma

2 2m \ ma) 2m

whence

^ sin 2 9

j(i+— +

(V ma ma t

p = \pacf/m 2 fl + — H— j-$) e J cos 6% .(41).

F r \ ma ra 2 a 2 / r* x

30 ON THE EFFECT OF THE INTERNAL FRICTION OF FLUIDS

17. The symbolical equations (40), (41) contain the solution
of the problem, the motion of the sphere being defined by the
symbolical equation (39). If we wish to exhibit the actual results
by means of real quantities alone, we have only to put the right-
hand members of equations (39), (40), (41) under the form
U+J — 1V, and reject the imaginary part. Putting for short¬
ness

/ n

V V

we have m = v (1 + J — l), and we obtain

P = - sin nt
n

.(43),

yfr=^a 2 c sin 2 9

3 \

1 4* 5 —- cos nt + * ,

2 vaj 2va \ va,

— (l + ” ) sin nt

2va

c-v(r-d)

cos (nt — vr + va)

+ (l + Aj s i a (ni — vr + va)

• (44),

p = — \ pacn j(l + ^5_) sin nt

.±(l
2 va V

1 + ^) cos ?i<j cos 0 . .(45).

The reader -will remark that the £, y]r, p of the present article
are not the same as the ^[r, p of the preceding. The latter are
the imaginary expressions, of which the real parts constitute the
former. It did not appear necessary to change the notation.

When fx = 0 , v = co , and ifr reduces itself to

a 3 c . 2 a ^ a 3 . „ _ dP
sin 0 cos nt, or ^ sm 0 .

2 r 2 r dt

In this case we get from (26)

B = a >

d% cos 0

di~V~’

0 - K §!^,

and Bdr -f %rdd is an exact differential d<f> where

, d|fcos 0
dt r 1 ’

= — % a 3

ON THE MOTION OF PENDULUMS.

31

which agrees with the result deduced directly from the ordinary
equations of hydrodynamics*.

18. Let us now form the expression for the resultant of the
pressures of the fluid on the several elements of the surface of the
sphere. Let P r be the normal, and T e the tangential, component
of the pressure at any point in the direction of a plane drawn
perpendicular to the radius vector. The formulae (4), (5) are
general, and therefore we may replace x } y in these formulae by
x , y\ where x , y' are measured in any two rectangular directions
we please. Let the plane of x y f pass through the axis of x and
the radius vector, and let the axis of x be inclined to that of x at
an angle which after differentiation is made equal to 6. Then
P v T z will become P r , T e> respectively. We have

u = R cos (6 — S-) — © sin {9 — ^),
v=R sin (6 - S-) + ® cos (0 - $•),

and when 6 = &

d _ cl d __ d
dx dr 9 dy rdd >

whence

dvl dE

du dR

©

dv __

dx dr *

II

^ i

i

r ’

dx

dR

Te = -i*>

(dR

d®_

\rdd

dr

d!0.

dr ;

(46).

In these formulae, suppose r put equal to a after differentiation.
Then P r , T e will be the components in the direction of r, 6 of the
pressure of the sphere on the fluid. The resolved part of these in
the direction of x is

P r cos 6 — T e sin 6,

which is equal and opposite to the component, in the direction of
x, of the pressure of the fluid on the sphere. Let F be the whole
force of the fluid on the sphere, which will evidently act along the
axis of x. Then, observing that 2nrad sin Odd is the area of an
elementary annulus of the surface of the sphere, we get

F = 27 m* f (- P r cos 6 + T e sin 8) a sin Odd .(47),

See Camb. Phil. Trans. Vol. vm. p. 119. [Ante, Vol. i. p. 41.]

32 ON THE EFFECT OF THE INTERNAL FRICTION OF FLUIDS

the suffix a denoting that r is supposed to have the value a in the
general expressions for P r and T e .

The expression for F may be greatly simplified, without em¬
ploying the solution of equations (27), (28), by combining these
equations in their original state with the equations of condition
(30). We have, in the first place, from (26)

_ 1 djr @ = __ 1 dty

r* sin 0 dO 9 r sin 0 dr

(48).

Now the equations (30) make known the values of ^ and d^/dr,
and of their differential coefficients of all orders with respect to g,
when r = a. When the expressions for R and © are substituted
in (46), the result will contain only one term in which the differen¬
tiation with respect to r rises to the second order. But we get
from (21), (27), (28) •

d 2 ty ___ sin 0 d / 1 1_ d\fr 2

dr 2 r 2 dd Vsin 6 d0) fi dt ’

and the second of equations (30) gives the value for r = a of the
first term in the right-hand member of the equation just written.
We obtain from (48) and (30)

= 0 ,

dR\ _ sing d% __ /0\

.rddJtT a dt~\r ) a *

(§®\ = _ __i_ (<th\

\ dr) a fi a sin 6 V dt ) a '

Substituting in (47), and writing p!p for pu, we get

F = 27 ra J | - ap a cos 0 + p | sin 0d0.

With respect to the first term in this expression, we get by
integration by parts

fp cos 0 sin 0 d0 = ^ sin 2 g. p — J / sin 2 0 ^ d0.

du

The first term vanishes at the limits. Substituting in the second

ON THE MOTION OF PENDULUMS.

33

term for dp/dd the expression got from (29), and putting r = a, we
get

/>• cos d ^ ed6 =-y jtll ($). sin 9 de -

Substituting in the expression for F\ we get

*-^4CKiFl +2 M si,ie ‘ w . w

19. The above expression for F, being derived from the
general equations (27), (28), combined with the equations of con¬
dition (30), holds good, not merely when the fluid is confined by a
spherical envelope, but whenever the motion is symmetrical about
an axis, and that, whether the motion of the sphere be or be not
expressed by a single circular function of the time. It might be
employed, for instance, in the case of a sphere oscillating in a
direction perpendicular to a fixed rigid plane.

When the fluid is either unconfined, or confined by a spherical
envelope concentric with the sphere in its position of equilibrium,
the functions fa, fa consist, as we have seen, of sin 2 0 multiplied
by two factors independent of 6. If we continue to employ the
symbolical expressions, which will be more convenient to work
with than the real expressions which might be derived from them,
we shall have

-1 »t£ ( r ) ( e V -1«</ 2 (r),

for these factors respectively. Substituting in (49), and perform¬
ing the integration with respect to 6 , we get

j = t J~1 {«/; (a) + 2/ 2 (a)} .(50).

20. Consider for the present only the case in which the fluid
is unlimited. The arbitrary constants which appear in equations
(38) were determined for this case in Art. 16. Substituting in
(50) we get

F = - * Won (1 + — + -JO e^"*.

3 r \ ma ma J

Putting for m its value 1/(1+^ —1), and denoting by M r the
S. m. 3

84 ON THE EFFECT OF THE INTERNAL FRICTION OF FLUIDS

mass of the fluid displaced by the sphere, which is equal to f 7rpa 3 ,
we get

F= - *'<» {(i + V-l + £ (l + i,)} i

whence

A

4<m

( 51 ).

Since J—l has been eliminated, this equation will remain un¬
changed when we pass from the symbolical to the real values of
F and

Let r be the time of oscillation from rest to rest, so that wt=7t,
and put for shortness h , Tc' for the coefficients of M f in (51); then

h = l+ i^’ ^ = i( 1 + m). (52) '

The first term in the expression for the force F has the same
effect as increasing the inertia of the sphere. To take account of
this term, it will be sufficient to conceive a mass MF collected at
the centre of the sphere, adding to its inertia without adding to
its weight. The main effect of the second term is to produce a
diminution in the arc of oscillation: its effect on the time of
oscillation would usually be quite insensible, and must in fact be
neglected for consistency’s sake, because the motion of the fluid
was determined by supposing the motion of the sphere permanent,
which is only allowable when we neglect the square of the rate of
decrease of the arc of oscillation.

If we form the equation of motion of the sphere, introducing
the force F , and then proceed to integrate the equation, we shall
obtain in the integral an exponential e~ st multiplying the circular
function, 8 being half the coefficient of d%/dt divided by that of
d 2 %/dtf. Let M be the mass of the sphere. My 2 its moment of
inertia about the axis of suspension, then

nVM' (l + a) 2 = 28 {My 2 + kM' (l + a) 2 }.

In considering the diminution of the arc of oscillation, we may
put l + a for 7. During i oscillations, let the arc of oscillation
be diminished in the ratio of A 0 to A., then

ON THE MOTION OF PENDULUMS.

35

For a given fluid and a given time of oscillation, both, h and ¥
increase as a decreases. Hence it follows from theory, that the
smaller be the sphere, its density being supposed given, the more
the time of oscillation is affected, and the more rapidly the arc of
oscillation diminishes, the alteration in the rate of diminution of
the arc due to an alteration in the radius of the sphere being more
conspicuous than the alteration in the time of oscillation.

21. Let us now suppose the fluid confined in a spherical
envelope. In this case, we have to determine the four arbitrary
constants which appear in (38) by the four equations (35) and (36).
We get, in the first place,

- + Be? + Oe~ na (l + —) + De ma (l-—) = Wc .(54),

-- + IBo?- Of™(ma+ 1 +—) + De m (ma-1 + —) = o*c (55),

a \ ma] \ ma)

i ++ i 1 + a) + (i - a) - 0 .«

-~ + 2 BP-Ge*' (mb +1 + i) + De mb (mS-l + ^)=0 (57).

Putting a 2 cK for af t '(a) + 2f 2 (a), which is the quantity that we
want to find, we get from (38) and (54)

■ ?.w

Eliminating in succession B from (54) and (55), from (56) and (57),
and from (54) and (56), we shall obtain for the determination of
A, C, D three equations which remain unchanged when a and b
are interchanged, and the signs of A , C y and D changed. Hence
— A, — C } — D are the same functions of b and a that A , G, D are
of a and b. It will also assist in the further elimination to observe
that G and D are interchanged when the sign of m is changed.
The result of the elimination is

K= 1-

3 b

7) (a, b) — r) (6, a)

2mV ’ 12 mal + £ (a, b) + f (6, a)
the functions £, 7) being defined by the equations

•(59),

7) (< a , b) = (mV 4* 3ma 4- 3) (mV — 3 mb 4- 3) e m(?J “ a)
f(a, b) ={b(m 2 b 2 -3mb+S)~-a(m 2 a 2 +3ma+S)}e m{h ~ a \

....(60).

3—2

36 ON THE EFFECT OF THE INTERNAL FRICTION OF FLUIDS

It turns out that K is a complicated function of m and
and the algebraical expressions for the quantities which answer
to h and Je in Art. 20 would be more complicated still, because
v (1 + V— 1 ) would have to be substituted for m in (60) and (59),
and then K reduced to the form — & 4- V — 11c'. To obtain nu¬
merical results from these formulae, it would be best to substitute
the numerical values of a , b, and v in (60) and (59), and perform
the reduction of K in figures.

22 . If the distance of the envelope from the surface of the
sphere be at all considerable, the exponential e v( - i ~ a \ which arises
from e m( - b ~ a \ will have so large a numerical value that we may
neglect the terms in the numerator and denominator of the fraction
in the expression for K which contain as well as the term

in the denominator which is free from exponentials, in comparison
with the terms which contain 6 v ( b ~ a \ Thus, if b- a be two inches,
Tone second, and \/fj! = m 116, we have €*'@-<0=2424000000, nearly;
and if b - a be only an inch or half an inch, we have still the
square or fourth root of the above quantity, that is, about 49234
or 222 , for the value of that exponential. Hence, in practical
cases, the above simplification may be made, which will cause the
exponentials to disappear from the expression for K We thus
get

V— i _ (mV + 3 ma + 3) (m 2 b* ~ 3 mb + 3) .

2 mV b (■ m 2 b 2 - 3 mb + 3) - a (mV + 3 ma + 3)' ’‘^ '*

If we assume

Zva + 3 + (2 z/V 4 - Sva) V— 1 = A (cos a + V- 1 sin a),

- Zvb 4- 3 + (2vV — 3z >b) V — 1 — B' (cos ft 4- V —1 sin ft),
bB f cos ft — a A cos a — O cos 7 ,
bB f sin ft — a A sin a = C' sin 7,

we get from (61)

jr 1 Zb^J-1 AB\

^ =1 + - • —qt {cos (a 4-/3 - 7 ) 4- V~1 sm (a + /3- 7 )},

whence

7 BbA'B' . , . . ,
A: = WC' sm(a + ^-' y )- 1

,, 3 bA'B' . a .

* = wa ,cos ^ +/3_ ^ ;

.( 62 );

OK THE MOTION OF PENDULUMS.

37

and, as before, hM' is the imaginary mass which we must conceive
to be collected at the centre of the sphere, in order to allow for
the inertia of the fluid, and — k'M'n dfydt the term in F on which
depends the diminution in the arc of oscillation.

If we suppose p! = 0, and therefore m = oo , we get from

b 5 + 2a 3
2(b 3 -a 3 )

(63),

and, in this case, k is the same as with AT sign changed, and &' = 0,
which agrees with the result obtained directly from the ordinary
equations of hydrodynamics* If, on the other hand, we make
b = oo , we arrive at the results already obtained in Art. 20. In
both these cases it becomes rigorously exact to neglect in the
expression for K —l given by (59) all the terms which are not
multiplied by 6 v(fi ~ a, \

If the effect of the envelope be but small, which will generally
be the case, it will be convenient to calculate k and k f from the
formula (52), which apply to the case in which b = oo, and then
add corrections A k, A k' due to the envelope. We get from (61)

AJc — V — 1A k' =

3 (mV + 3 ma + 3) 2 , v

2 m 2 a 5(mV-3m5+3)-a(mV+3ma+3)**^ bJ )y

which may be treated, if required, as the equation (61) was treated
in the preceding article. If, however, we suppose m large, and are
content to retain only the most important term in (64), we get
simply

»-W=Vi> M '-° . (65) ’

so that the correction for the envelope may be calculated as if the
fluid were destitute of friction.

See Camb. Phil. Trans. Vol. vm. p. 120. [Ante, Vol. i. p. 41.]

UN JL-tUli J1AJ1 UJ3 1 -EUIj 1JN 1 JiixvlN-cxJu JD JLU.UXJLU1N UJ? JCLiUiJJO

Section III.

JU on of the equations in the case of an infinite cylinder oscillating
in an unlimited mass of fluid, in a direction perpendicular to
‘ l s axis.

oppose a long cylindrical rod suspended at a point in its
ade to oscillate as a pendulum in an unlimited mass of
resistance experienced by any element of the cylinder
etween two parallel planes drawn perpendicular to the
anifestly be very nearly the same as if the element
3 an infinite cylinder oscillating with the same linear
For an element situated very near either extremity of
uhe resistance thus determined would, no doubt, be sensibly
ux^ueous; but as the diameter of the rod is supposed to be but
small in comparison with its length, it will be easily seen that the
error thus introduced must be extremely small.

Imagine then an infinite cylinder to oscillate in a fluid, in a
direction perpendicular to its axis, so that the motion takes place
in two dimensions, and let it be required to determine the motion
of the fluid. The mode of solution of this problem will require
no explanation, being identical in principle with that which has
been already adopted in the case of a sphere. In the present
instance the problem will be found somewhat easier, up to the
formation of the equations analogous to (33) and (34), after which
it will become much more difficult.

25. Let a plane drawn perpendicular to the axis of the
cylinder be taken for the plane of xy , the origin being situated in
the mean position of the axis of the cylinder, and the axis of x
being measured in the direction of the cylinder’s motion. The
general equations (2), (3) become in this case

dp _ fdhi d 2 u\ du 1

dx ^ \dx 2 dy 2 ) ^ dt I

dp _ (d 2 v d 2 v\ dv

dy ^ \dx 2 "** dy 2 ) ^ dt .

( 66 ),

du dv
dx " r dy

( 67 ).

ON THE MOTION OF PENDULUMS.

39

By virtue of (67), udy — vda> is an exact differential. Let tlien
vdy — vdx = dx .(68).

Eliminating p by differentiation from the two equations (66),
and expressing u and v in terms of % in the resulting equation, we
get

f d 2 d 2 1 d\ f d 2 d 2 \ A

W + ~ p' It) W + 3j?) X ~ °. (69 ^’

and, as before

X = Xi + %2.(70),

where

/ d 2 d 2 \ A /rri x

w + ^) %i_0 . W’

/ cP d 2 1 d\ A

(M + ( ' 72 ' 1 '

We get from (66) and (68)

, , j d f d 2 d 2 1 d\

dp-fipdx.

, , d (cP d 2 1 d\

y * dx \dx 2 dy 2 y! dt) ^

which becomes by means of (70), (71), and (72)

. (73> -

26. Passing to polar co-ordinates r, 8 , where 9 is supposed to
be measured from the axis of x, we get from (68), (71), (72), and

(73)

Brd& — ®dr = dx .(74),

/d* 1 d 1 d?\

Ur' 2 + r dr + r i d0‘J Xl ~° . ^’

(d 2 Id Id 2 1 d\ __ .

V^r 2 r dr~^~ r 2 dd 2 \j! dt) .^ ’

dp= Plt&r rd6 -% dr ) .

R, © in (74) being the velocities along and perpendicular to the
radius vector.

40 ON THE EFFECT OF THE INTERNAL FRICTION OF FLUIDS

27. Let a be the radius of the cylinder; and as before let the
cylinder’s motion be defined by the equation

d l

dt

= ce^ zrin ' t = ce 1 ^' 7

(78);

then we have for the equations of condition which relate to the
surface of the cylinder

B = ntd- C0 * d S =

© = ■

dr dt

o cos 6

c sin 8

\ whenr=a...(79).

The general equations (75), (76), as well as the equations of
condition (79), may be satisfied by taking

Xi = &' mH sin 0 F t (r), = €» mH sin 0 F 2 (r).(80).

Substituting in (75), (76), and (79), we get

+ 0.(81),

K (r) +1 F; (r) - I F, (r) - m*F, (r) = 0.(82),

F x (a) + F t (a) = ac, F; (a) + F; (a) = c .(83),

besides which we have the condition that the velocity shall vanish
at an infinite distance.

28. The integral of (81) is

F x( r ) = ~ + Fr .(84).

The integral of (82) cannot be obtained in finite terms.

To simplify the latter equation, assume F 2 (r) = F' (r). Sub¬
stituting in (82), and integrating once, we get

K ( r ) + b,'(f)- m?F s (r) = 0.(85).

It is unnecessary to add an arbitrary constant, because such
constant, if introduced, might be got rid of by writing F M +
for F s (r).

o &

ON THE MOTION OF PENDULUMS.

41

To integrate (85) by series according to ascending powers of r,
let us first, instead of (85), take the equation formed from it by
multiplying the second term by 1 — S. Assuming in this new
equation F 3 (r) = A l aP + B J aP +..., and determining the arbitrary
indices a, /3... and the arbitrary constants A /} B r .. so as to satisfy
the equation, we get

F 3 (r) = A,

mV mV

2(2-8) + 2.4(2 — S) _ (4 — S)

+ ..

+ A +

mV mV

2(2 + 8) + 2.4 (2 + 8) (4 + 8) +

= ( A , + A „ + A J io g r )

1 +

mV

mV
+ 2 r T4 5

+

+ i A- A) « S, + ££ 8, + ^A + •

4- terms involving 8 2 , S 3 —

In this expression

5f i = r 1 + 2“ 1 -l-3~ 1 ...+r 1 .(86).

Putting now

substituting in the above equation, and then making S vanish, we
get

K (r) = (G+D log r) (l + ^ + ...)

~D

2 ~ mV ~ mV q

^1 ^ g2 ^2 02 ^3 "b • • •

•(87).

The series in this equation are evidently convergent for all values
of r, however great; but, nevertheless, they give us no information
as to what becomes of F 3 (r) when r becomes infinite, and yet one
relation between G and D has to be determined by the condition
that F b (r) shall not become infinite with r.

The equation (85) may be integrated by means of descending
series combined with exponentials, by assuming

F 3 ( r ) = e ±mr (A/r a + BpP ...).

I have already given the integral in this form in a paper, On the

42 ON THE EFFECT OF THE INTERNAL FRICTION OF FLUIDS

numerical calculation of a class of definite integrals and infinite
series *. The result is

F t (?)=Gi

r.3 2

+jy e mr T- i \ 1 +

l 2 .3 2 .5 2

2.4mr + 2.4 (4mr) 2 2.4.6 (4 mrf

l 2 l 2 .3 2 l 2 .3 2 .5 2

+

J_

2.4mr 2.4 (4mr) 2 1 2.4,6 (4mr) 8

-I

...( 88 ).

These series, although ultimately divergent in all cases, are
very convenient for numerical calculation when the modulus of mr
is large. Moreover they give at once B' — 0 for the condition that
Fjf) shall not become infinite with r, and therefore we shall be
able to obtain the required relation between C and D, provided we
can express D' as a function of G and D.

29. This may be effected by means of the integral of (85)
expressed by different integrals. This form of the integral is
already known. It becomes, by a slight transformation,

P 3 (r) = P [G" + D" log (r sin 2 go)} (e mr cosw + e - mrcos<a ) dco.. .(89),

Jo

C", D" being the two arbitrary constants. If we expand the
exponentials in (89), and integrate the terms separately, we obtain,
in fact, an expression of the same form as (87). This transforma¬
tion requires the reduction of the definite integral

7T

Pi = f cos 2 * co log sin co dco.
Jo

If we integrate by parts, integrating cos co log sin co dco , and differ¬
entiating cos 2i-1 w, we shall make P. depend on P.^. Assuming
P„ = Qo, Pi = iQv • •» and generally

1.3...(2^-1)
2.4...2i

Qt,

we get

Qi=Q- + 4- 1 .. .+ (2»n ~ = l log ( 1 ) - 18 t f.

* See Gamb. Phil. Trans. Vol. ix. p. 182. [Ante, Vol. n. p. 849.]
t A demonstration by Mr Ellis of the theorem

/,

2 log sin 0 d0 = - log (£),

ON THE MOTION OF PENDULUMS.

43

The equivalence of the expressions (87) and (89) having been
ascertained, in order to find the relations between C 3 D and C", D’\
it will be sufficient to write down the two leading terms in (87)
and (89), and equate the results. We thus get

0 + D log r = 7 t C" + ttD' log r + 2 t tD" log (J),

whence

G = ttC" + 2 , jt log (J). D", D = ttZ)".(90).

There remains the more difficult step of finding the relation
between D' and G", D". For this purpose let us seek the ultimate
value of the second member of equation (89) when r increases
indefinitely. In the first place we may observe that if XI, Cl' be
two imaginary quantities having their real parts positive, if the
real part of Cl be greater than that of Cl', and if r be supposed to
increase indefinitely, e nr will ultimately be incomparably greater
than € a ' r , or even than logr.e ov , or, to speak more precisely, the
modulus of the former expression will ultimately be incomparably
greater than the modulus of either of the latter. Hence, in finding
the ultimate value of the expression for F s (r) in (89), we may
replace the limits 0 and \ir of co by 0 and a> lt where co 1 is a positive
quantity as small as we please, which we may suppose to vanish
after r has become infinite. We may also, for the same reason,
omit the second of the exponentials. Let cos co = 1 — A, so that

. o - /- A\ 7 d\ /_ \ \ d\

<h “jw=t --{ 1 + i + -)jip:y

then the limits of A, will be 0 and A x , where \ = 1 — cos co r Since
log ^ ultimately vanishes, and 1 -f ^ + ... becomes ultimately
1, we get from (89)

limit of F & (r) = e mr

x limit of p(0" + D" log 2\r) e~ nXr *

due to Euler, will be found in the second volume of the Cambridge Mathematical
Journal , p. 282, or in Gregory’s Examples , p. 484.

* The word limit is here used in the sense in which /(r) may be called the limit
of 0 (r) when the ratio of 0 (r) to f(r) is ultimately a ratio of equality, though / (r)
and 0 (r) may vanish or become infinite together, in which case the limit of 0(r),
according to the usual sense of the word limit, would be said to be zero or
infinity.

44 ON THE EFFECT OF THE INTERNAL FRICTION OF FLUIDS

If now we put X = XV *, we shall have 0 and X x r for the limits of
X', and the second of these becomes infinite with r. Hence

limit of FJr) = (2 r)~* e mr [ (G" + D" log 2V) e -"*' V - * d\' .(91).

J 0

Now e~ x x~* dx = 7 r^, and if we differentiate both sides of the
J o

equation

f e- X s " 1 dx = T(s)

J o

with respect to s, and after differentiation put s = we get

[ 6~~ x x“"^log x dx = F (|).

J o

Putting x = mX' in these equations we get

f e~' mX ' X'“* dA/ = 7 rhn~%,

Jo

f 6~ my X'~* log X' dX' = {F (|) — 7r£ log m},

Jo

where that value of is to be taken which has its real part
positive. Substituting in (91) we get

limit of FJr) = e"" { 0" + FJ - log f) £"}.

Comparing with (88) we get

M£n c " + (*'* r ' i - lo ei) i) "}. (92) '

30. We are now enabled to find the relation between G and
D arising from the condition that the motion of the fluid shall not
become infinitely great at an infinite distance from the cylinder.
The determination of the arbitrary constants A , B, G , D will
present no further difficulty. We must have B = 0, since other¬
wise the velocity would be finite at an infinite distance, and then
the two equations (83), combined with the relation above men¬
tioned, will serve to determine A , (7, D. The motion of the fluid
will thus be completely determined, the functions F x (r\ F z (r)
being given by (84) and (87). When the modulus of mr is large,
the series in (87), though ultimately hypergeometrically conver-

ON THE MOTION OF PENDULUMS.

45

gent, are at first rapidly divergent, and in calculating the numerical
value of F Q (r) in such a case it would be far more convenient to
employ equation (88). The employment of this equation for the
purpose would require the previous determination of the constant
(7. It will be found however that in calculating the resultant
pressure of the fluid on the cylinder, which it is the main object
of the present investigation to determine, a knowledge of the value
of O' will not be required, and that, even though the equation (88)
be employed*.

Putting D' = 0 in (92), and eliminating G " and D” between the
resulting equation and the two equations (90), we get

C=(log|-7r-ir'l)i>.(93);

and we get from (88) and (84), observing that F 2 ( r ) = F 3 (r), and
that B = 0,

f + F s( a ) = ac ’ - f + aF s" ( a ) — ac .(94),

(If (Ju

whence

a 2 c + A aF 3 (a)

a?c-A~ F'{a) . ^ s>)m

This equation will determine A , because if F 3 (a) be expressed by
(87) the second member of (95) will only contain the ratio of G to
D, which is given by (93), and if F 3 (a) be expressed by (88) G'
will disappear, inasmuch as D ' = 0.

31. Let us now form the expression for the resultant of the
forces which the fluid exerts on the cylinder. Let F be the
resultant of the pressures acting on a length dl of the cylinder,
which will evidently be a force acting in the direction of the axis
of x; then we get in the same way as the expression (47) was
obtained

F= adl (— P r cos 6 + T e sin 9) a d6 .(96),

J o

and P r , T e are given in terms of R and © by the same formulae
(46) as before. When the right-hand members of these equations
are expressed in terms of there will be only one term in which

* [C as subsequently determined will be given at the end of the paper.]

46 ON THE EFFECT OF THE INTERNAL FRICTION OF FLUIDS

the differentiation with respect to r rises to the second order, and
we get from (70), (75), and (76)

,2 r dr r 2 d6 2 /i dt

dr

We get from this equation and the equations of condition (79)

(dft\ = 1 (ix\ _ l ( d \ \ = n

\drj a a \dd) a a 2 \drd61 a ’

( 2®. \ — 2. _sin 6 dg_ ®

\rdd) a ~ a 2 \d6 2 ) a ~ a dt~ a ’

(fx\ = 1

dr/a

\dF) a a Wr) a + a 2 ($ 2 ) 0 fi'i,dt) a fi'[dt)„

Hence

F = adlJ j— p a cos 0 + p (^|0 sin#jd#. (97).

We get by integration by parts

Jp a cos 0 dd = p a sin 6 - j [ sin 6 dd.

The first term vanishes at both limits; and putting for dp/dd its
value given by (77), and substituting in (97), we get

F=padi itl 0 Hii +( W sin ^

or

F= irpadl . n J - 1 {aF/ (a) + F 8 ' (a)}

Observing that F 8 '(a) or F 2 (a) =ac - F x (a) from (83), and that
F^a) = Aa \ where A is given by (95), and putting M for nrpcfdl,
the mass of the fluid displaced, we get

F = Men J~ 1 (l - 2 ~ e^ru

v 1 aF^(a) + F s '(d)) u

which becomes by means of the differential equation (85) which F
satisfies 3

Let

F-McnJ^ . (98>

= k-J^lk' .( 99 ),

1-

WJ («) .

m 2 aF 3 (a )"

ON THE MOTION OF PENDULUMS.

47

where k and k' are real, then, as before, kM' cfg/df will be the part
of F which alters the time of oscillation, and k'M'nd^jdt the part
which produces a diminution in the arc of oscillation.

When fx! vanishes, m becomes infinite, and we get from (88)
and (99), remembering that Zf = 0; & = 1, &' = (), a result which
follows directly and very simply from the ordinary equations of
hydrodynamics *

32. Every thing is now reduced to the numerical calculation
of the quantities k, k', of which the analytical expressions are
given. The series (87) being always convergent might he employed
in all cases, but when the modulus of ma is large, it will be far
more convenient to employ a series according to descending powers
of a. Let us consider the ascending series first.

Let 2tll be the modulus of ma ; then

_ jV“i, a /n a / nr ^

*“2vr»VA.< ioo >>

r being as before the time of oscillation from rest to rest. Sub¬
stituting in (99) the above expression for ma, we get

Jr J i ]»' ~ i i ^ ~~ ^ .flOl).

k 7 lk-l+ m t F ^ 1 '

Putting for shortness

log e 4 + ^- J r'ft) = -A .(102),

we get from (87) and (93)

1 j ( a ) = ( lo g m+A + ^-v/-!

D J

i + p 7 - i - lS ^ S is/ ~ + "7

- ^ S. - lT D ? S. V-i + ...),

1 m 2 ,_ m 4

ctF; (a) = 1 + -p \/ — 1 — p—2^ — • - •

+ 2 m + A + J /-I) l - -

- 2 - «L 8, - pJ^-5 S.J=i + ...) .

-)

* See Camb. Phil. Trans . Vol. vm. p. 116. [Ante, Yol. i. p. 37.]

48 ON THE EFFECT

OF THE INTERNAL FRICTION OF FLUIDS

Let

m' 2

in 6

nr

m 8

l 2 .2 2 .3

+ ... = # 0 , jPg - ~ Me

m 6

nr

l 2 l 2 .2 2 .3 !

tn

m

nr

71/T / 111

, 4 -... = Jmq } 2 2 2 2 3 2 4 5

,+...=iiL:

yA - s t +... =A, ^

m 2 m 6 r, , r , m 4 „ tn 8 „ , -w
^ “ 2 y g 3 + ...=N 0 , b “ + ''' ~ A f j

log e m + A =L .0 () 4):

then substituting in (101), changing the sign of J - l, and
arranging the terms, we get

Jc + J^lk'

(103),

= 1 +

m 2

ZM'-\m;+n.+[ lM 0 +LM-l (1 -.)//) - .V„p-
-\m;+l (1 -MD+N.'+i-LM,'-? (1 -il/,0 + AVJ J- 1

2 -I>K+

33. Before going on with the calculation, it will Ik* ivquisitr
to know the numerical value of the transcendental quantify A.
Now

*~ l r (i) = (r*r r (i) = ^ log r («)

^.,« r 0f„r.,

and the value of d/ds.\ogV(l + 5 ) may be got at once fmni
Legendre's table of the common logarithms of F (I q s), in u hich
the interval of s is ’001. Putting l s for the tabular number corn*
sponding to s, we have

J s log r (1 + s) = 1000 log e 10 {AZ, - \ AH, + < AH, - } A 7 ( ... |.
For 3 = | -

Al, = +16050324, A 2 il s = + 405020, A% = -35!>, AH, t (1*,

* These numbers are copied from De Morgan’s Diffarndul „,„i r„l n ,.

his , p. 588.

ON THE MOTION OF PENDULUMS.

49

the last figure being in each case in the 12th place of decimals.
We thus get

t r-*T (-1) = -1*9635102, A = + *5772158*.(106).

34. When in is large, it will be more convenient to employ
series according to descending powers of a. Observing that the
general term of F z (a) as given by (88), in which D' = 0, is

,_ ivr [i.3...( 2t-i )r
^ ' 2.4...2i (imcCfafr 5

we get for the general term of F z (a)

file's" L(^-l)* 2*'- 1

^ ^ 2.4... (2i — 2) (4 maf" 1 \ 2%. 4 ma 2a

and the expression within brackets is equivalent to

(2 i-l)(2i+l)

whence

aF^a) = CV"" 1 ma*

Sia

L*JL A:. 3 A

2.4ma + 2.4 (4wict) 2

and we find by actual division

A/(A

; — ma — ^ 4- J, (ma) 1 _

85. When many terms are required, the calculation of the
cooilieienls may be facilitated in the following manner.

Assuming aF,' (a) = v (a) F z (a), we have
F' (a) =■ a 1 v (a) l<\ (a.),

/' T a " (a) - [u7 l v (a) — aF 2 v {a) + a 2 (va) 2 } F {) (a).

Substituting in the differential equation (85) which A 3 has to
satisfy, we get-

av (a) [v (a,)} 2 — m\c = 0 .(107).

Assuming

v (a) — — ma 4- A 0 4~ A x (ma) 1 ~f A 2 (ma) *4-. (108),

* [A is in fact the well-known transcendent nailed Iiitler'a Constant, Urn value, of
which is *r»77‘21TUU; UJ A-e. This, which 1 oii"lit to lutvci known, was pointed out to
me just after the publication of the paper by my Iriend Urol. 1 ( . Newman. |

S. III.

4

OF THE INTERNAL FRICTION OF FLUIDS

50 ON THE EFFECT

and substituting in the above equation, we get

- mo - 1A W 1 ~ 2 A (“)■“ - 3 A ( ma Y a ■ • •

- +{-2ma + A+A( ma )" 1+ "-HA + A( ma ) 1 + ---} ==0 >

which gives on equating coefficients, A 0 = — £, and foi i > 0
2 A m = — iA, + A 0 A. + AjA ^... + A { A 0 ,

or, assuming to avoid fractions,

A=2^B t .OOi)),

B,., = - 2 iB ( + Bfi, + B t B„ ...+B i B a .(110),

a formula by means of which the coefficients B t , B,> B a ... may be*
readily calculated one after another. We get

B,=- 1, A = + !. A = J? 3 == + 25,'

B=- 208, B s = + 2146, B e = - 20:168,
i? 7 = + 375733, £ 8 = - 6092032.

(111).

We get now from (100), (101), (108), and (109)

h _ V - l ]d

= l + 2e ^nT-lV^

XXf

( 112 ),

whence if we calculate

m,= 2m' 1 , w 2 = - £ B 0 var\ u s = //.m' 1 ...,

u t = (- l ) i+1 1 B { _ 2 8~ m m - ',

we shall have, changing the sign of V—L in (112), and writ in”
8 for 1 ,

&+«/—1 &' = 1 + «,8+MjB 8 —m,8 3 +w 4 8 4 —w. 8 S s +... 1

i+« 2 - V f« 8 +VI m 6 — +\/ i 'q—V! w u • • • 1

• If 4*** I* e the common logarithms of Urn ooHliciunt.; o|
nr 1 , HI" 2 ... in the last two of the formula* (113),

l,= 1505150;
4 = 1-6989700;
4=2-6453650;

4=2-4948500;
4 = 2-2371251;
4=2-4046734;

4 = 2-3646348 ;
4 = 2-7019316 ;
4= 2-601701:. ;

ON THE MOTION OF PENDULUMS.

51

and if the logarithms of the coefficients of Wf 1 , in u v u 2 ...

be required, it will be sufficient to add ‘1505150 to the 1st, 3rd,
5th, &c. of the logarithms above given.

36. It will be found that when ttt is at all large, the series
(113) are at first convergent, and afterwards divergent, and in
passing from convergent to divergent the quantities become
nearly equal for several successive terms. If after having calcu¬
lated i terms of the first of the series (113) we wish to complete
the series by a formula involving the differences of u v we have
^ < -^ 1 8 m + ^8 H2 -...=tf{l-«(l + A )+ 8a (1+A) 2 -...}^

= 8* {1 + 8 (1 + A)} 1

and

7T . —-

rrr - -— . 7T ^ IT fl V “1

1 4- 8 = 1 + cos T + a/ — 1 sm j = 2 cos e 8 ,

4 4 a

SCl + sr^secI

so that the quantities to be added to 7c } 7c, arc

•j . i \ -i i 7r f 2i — 1 , 7 r 2 i A

to at, (— 1) }, see t) - cos 0 tt . u. — J sec ». cos - 7r. Am.

8 1 ~ 8 8 £

/- tt\ 2 2i+l a2 |

+ (isecgj cos 8 tt.A 7/ v ..V

; V(U4).

, ,, , -w , 7T f . 2'£ —1 7T . 2i .

to /»:, (- IV A sec , ism t) 7r . m. - J sec ., sm - i r. A/c
v ' - 8 ( 8 £ ~ 8 8

/ 7rV 2 . 2?/ 4- 1 . )

+ f i sec -J sm - 8 - 7r. A ?/^... r

37. Tlio following table contains the values of the functions
and 1c calculated for 40 different valiums of 1U. From 1U = * I to
lU = 1*5 flu; calculation was performed by means of flu; formula,
(105) ; tint rest of the fable; was calculated by means of the series
(113). In the former part of the calculation, six places of deci¬
mals were employed in calculating the functions i/ () , &o. given
by (103). The last figure was then struck out, and five-figure
logarithms were employed in multiplying the four fund,ions df 0 ,

4—2

52 ON THE EFFECT OF THE INTERNAL FRICTION OF FLUIDS

M e , and 1 - M' e by ir/% and by L, as well as in reducing tbe
right-band member of (105) to the form k + J — Ik’. The terms
of the series (113) were calculated to five places of decimals. That
these series are sufficiently convergent to be employed when
ttt —1*5, might be presumed from the numerical values of the
terms, and is confirmed by finding that they give k = 1*952, and
&' = ri53. For ttt = 1*5 and a few of the succeeding values, the
second and third of the series (113) were summed directly as far
as in’" 5 inclusively, and the remainders were calculated from the
formulae (114). Two columns are annexed, which give the values
of ttt% and ttl 2 &', and exhibit the law of the variation of the two
parts of the force F } when the radius of the cylinder varies, the
nature of the fluid and time of oscillation remaining unchanged.
Four significant figures are retained in all the results.

m

h

V

m 2 7 c

m 2 k '

tit

h

h '

m 2 /c'

0

00

00

0

0

2-1

1-677

•7822

7-395

3-450

•i

19*70

48-63

•1970

•4863

2-2

1-646

•7421

7-966

3-592

•2

9-166

16-73

*3666

•6691

2-3

1-618

•7059

8-557

3-734

*3

6-166

9*258

*5549

•8332

2*4

1-592

•6780

9168

3-877

•4

4-771

6-185

•7633

•9896

2-5

1-568

■6430

9-799

4-019

•5

3-968

4-567

*9920

1-142

2-6

1-546

•6154

10-45

4-160

*6

3-445

3-589

1-240

1-292

2-7

1-526

•5902

11-12

4-303

*7

3-082

2-936

1*510

1-439

2-8

1-507

•5669

11-81

4*444

*8

2-812

2-477

1-800

1-585

2-9

1-489

•5453

12-52

4-586

*9

2-604

2-137

2-110

1-731

3-0

1-473

•5253

13-25

4-728

1-0

2-439

1-876

2-439

1-876

3-1

1-457

•5068

14-01

4-870

1*1

2-306

1-678

2*790

2*021

3-2

1-443

•4895

14-78

5-012

1-2

2-194

1-503

3-160

2-164

3-3

1-430

•4732

15-57

5-154

1-3

2-102

1-365

3-552

2-307

3-4

1-417

•4581

16-38

5-296

1*4

2-021

1-250

3-961

2-450

3-5

1-405

•4439

17-21

5-437

1*5

1-951

1-153

4-389

2-595

3-6

1-394

•4305

18-06

5-580

1-6

1-891

1-069

4-841

2-739

3-7

1-383

•4179

18-93

5-721

1*7

1-838

•9965

5-312

2-880

3-8

1-373

•4060

19-82

5-863

1-8

1-791

•9332

5*804

3-024

3-9

1-363

•3948

20-73

6-005

1*9

1-749

•8767

6*314

3*165

4-0

1-354

•3841

21-67

6-145

2*0

1-711

•8268

6-845

3-307

00

1

0

00

00

The numerical calculation by means of the formulae (103),
(104), (105) becomes very laborious when many values of the
functions are required. The difficulty of the calculation increases
with the value of III for two reasons, first, the calculation of the
functions M oi &c. becomes longer, and secondly, the moduli of the
numerator and denominator of the fraction in the right-hand

ON THE MOTION OF PENDULUMS.

53

member of (105) go on decreasing, so that greater and greater
accuracy is required in the calculation of the functions M 0 , &c.,
and of the products LM 0 , &c., in order to ensure a given degree of
accuracy in the result. The calculation by the descending series
(113) is on the contrary very easy.

It will be found that the first differences of ttt 2 &' and of
m 2 (&-l) are nearly constant, except near the very beginning of
the table. Hence in the earlier part of the table the value of h or
Jc for a value of ttt not found in the table will be best got by
finding m 2 h — ttt 2 or ttt 2 k f by interpolation, and thence passing to
the value of Jc or Jc'. Very near the beginning of the table, inter¬
polation would not succeed, but in such a case recourse may be
had to the formulae (103), (104), (105), the calculation of which is
comparatively easy when ttt is small. It did not seem worth while
to extend the table beyond ttt = 4, because where ttt is greater
than 4, the series (113) are so rapidly convergent that Jc and Tc
may be calculated to a sufficient degree of accuracy with extreme
facility.

38. Let us now examine the progress of the functions Tc
and k'.

When ttt is very small, we may neglect the powers of ttt in the
numerator and denominator of the fraction in the right-hand
member of equation (105), retaining only the logarithms and the
constant terms. We thus get

* + \/-lA' = l-

whence

tnrW-i

L — \tt V — 1

m 2 (Jfe-

• d = —iz:_

} L*+ (&)•’

m 2 k’ =

-L

..(115),

X 2 + (£7 r ) 2

L being given by (102) and (104), or (104) and (10G). When ttt
vanishes, L, which involves the logarithm of ttt" 1 , becomes infinite,
but ultimately increases more slowly than if it varied as ttt affected
with any negative index however small. Hence it appears from
(115), that Jc — 1 and Jc are expressed by m" 2 multiplied by two
functions of ttt which, though they ultimately vanish with ttt,
decrease very slowly, so that a considerable change in ttt makes but
a small change in these functions. Now when the radius a of the
cylinder varies, everything else remaining the same, ttt varies as a,

54* ON THE EFFECT OF THE INTERNAL FRICTION OF FLUIDS

and in general the parts of the force F on which depend the altera¬
tion in the time of vibration, and the diminution in the arc of
oscillation, vary as « 2 &, cfk’, respectively. Hence in the case of a
cylinder of small radius, such as the wire used to support a sphere
in a pendulum experiment, a considerable change in the radius of
the cylinder produces a comparatively small change in the part of
the alteration in the time and arc of vibration which is due to the
resistance experienced by the wire. The simple formulae (115)
are accurate enough for the fine wires usually employed in such
experiments if the theory itself be applicable; but reasons will
presently be given for regarding the application of the theory to
such fine wires as extremely questionable.

From m = *3 or *4 to the end of the table, the first differences
of each of the functions m 2 (Jc — 1) and nr&' remain nearly constant.
Hence for a considerable range of values of m, each of the func¬
tions may be expressed pretty accurately by A + J?m. When m is
at all large, the first two terms in the 2nd and 3rd of the formulae
(113) will give 1c and 1c with considerable accuracy, because, inde¬
pendently of the decrease of the successive quantities tlT 1 , m~ 2 ,
nr 3 ..., the coefficients of m" 1 and m“ 2 are considerably larger than
those of several of the succeeding powers. If we neglect in these
formula! the terms after we get

k = i + </2. nr 1 , 1c = vs. m" 1 +1 m~ 2 .

It maybe remarked that these approximate expressions, regarded
as functions of the radius a, have precisely the same form as the
exact expressions obtained for a sphere, the coefficients only being
different.

ON THE MOTION OF PENDULUMS.

55

Section IV.

Determination of the motion of a fluid about a sphere which moves
uniformly with a small velocity . Justification of the applica¬
tion of the solutions obtained in Sections II. and III. to cases
in which the eoctent of oscillation is not small in comparison
with the radius of the. sphere or cylinder. Discussion of a
difficulty which presents itself with reference to the uniform
motion of a cylinder in a fluid.

39. Let a sphere move in a fluid with a uniform velocity V,
its centre moving in a right line ; and let the rest of the notation
be the same as in Section II. Conceive a velocity equal and
opposite to that of the sphere impressed both on the sphere and
on the fluid, which will not affect the relative motion of the sphere
and fluid, and will reduce the determination of the motion of the
fluid to a problem of steady motion. Then we have for the equa¬

tions of condition

B = 0, © = 0, when r = a .(116);

R — — V cos 6, © = Fsin 9, when r = oc .(117).

The equations of condition, as well as the equations of motion,
may be satisfied by supposing yjr to have the form sin 2 9f(r). We
get from (20'), by the same process as that by which (33), (34) were
obtained,

.ms),

the only difference being that in the present case the equation
(20') cannot be replaced by the two (22), (23), which become
identical, inasmuch as the velocity of the fluid is independent of
the time.

The integral of (118) is

f(r) = Ar" 1 + Br + Or 2 + Dr A .

which gives

B = As = 2 cos 6 ( Ar “ 8 + Br* + C + Br 2 ),

r sm 0 dd v

© =-J—r = sin 6 (. Ar " 3 4 Br 2 ).

r sind dr

(U9),

56 ON THE EFFECT OF THE INTERNAL FRICTION OF FLUIDS

The first of the equations of condition (117) requires that

D=0, C = -\V .(120).

It is particularly to be remarked that inasmuch as the two arbitrary
constants C, D are determined by the first of the conditions (117),
none remain whereby to satisfy the second. Nevertheless it
happens that the second of these conditions leads to precisely
the same equations (120) as the first. The equations of condition
(116) give

A=-iVa 3 , I? = f Va;

whence

, , 2 t 1 3 r

■'Jr = i Va (-j H--

r 4 ' a a

R

3a a'
'Yr + ¥r s

cos 8

~ r C

sin 2 6* .(121),

.( 122 ),

.(123).

If now we wish to obtain the solution of the problem in its
original shape, in which the sphere is in motion and the fluid at
rest, except so far as it is disturbed by the sphere, we have merely
to add V cos 8 , — V sin 8, \ Vr 2 sin 2 8 to the expressions for
R } ©, We get from (121)

t = i Va 2

3 r

(124).

40. Let us now return to the problem of Section II.; let us
suppose the time of oscillation to increase indefinitely, and examine
what equation (40) becomes in the limit.

When r becomes infinite, n , and therefore m, vanishes; the
expression within brackets in (40) takes the form oo — oo, and its
limiting value is easily found by the ordinary methods. We must
retain the m 2 in the coefficient of t, because t is susceptible of
unlimited increase. We get in the limit

= £ a 2 ce^' mH sin 2 9 .(125).

* I have already had occasion, in treating of another subject, to publish the
solution expressed by this equation, which I had obtained as a limiting case of the
problem of a ball pendulum. See Philosophical Magazine for May, 1848, p. 343.
[Ante, Yol. n. p. 10.]

ON THE MOTION OF PENDULUMS.

57

If now we put V for dg/dt, the velocity of the sphere, we get
from (39), c6*' mH =V. After substituting in (125), the equation
will remain unchanged when we pass from the symbolical to the
real values of ^ and V } and thus (125) will be reduced to (124).

41. It appears then that by supposing the rate of alteration
of the velocity of the sphere to decrease indefinitely, we obtain
from the solution of the problem of Section II. the same result as
was obtained in Art. 39, by treating the motion as steady. As yet,
however, the method of Art. 40 is subject to a limitation from
which that of Art. 39 is free. In obtaining equation (40), it was
supposed that the maximum excursion of the sphere was small in
comparison with its radius. Retaining this restriction while we
suppose t to become very large, we are obliged to suppose c to
become very small, so that the velocity of the sphere is not merely
so small that we may neglect terms depending upon its square, a
restriction to which Art. 39 is also subject, but so extremely small
that the space passed over by the sphere in even a long time is
small in comparison with its radius.

We have seen, however, that on supposing r very large in (40)
we obtain a result identical with (124), not merely a result with
which (124) becomes identical when the restriction above men¬
tioned is introduced. This leads to the supposition that the
solution expressed by (40) is in fact more general than would
appear from the way in which it was obtained. That such is
really the case may 'be shewn by a slight modification of the
analysis. Instead of referring the fluid to axes fixed in space, refer
it to axes originating at the centre of the sphere, and moveable
with it. In the general equations of motion, the terms which
contain differential coefficients taken with respect to the co¬
ordinates will remain unchanged, inasmuch as they represent the
very same limiting ratios as before: it is only those in which
differentiation with respect to t occurs that will be altered. If
d'/dt be the symbol of differentiation with respect to t when the
fluid is referred to the moveable axes, we shall have

d __ d' d% d

dt dt dt dx 9

but the terms arising from d%/dt . d/dx are of the order of the square
of the velocity, and are therefore to be neglected. Hence the

58 ON THE EFFECT OF THE INTERNAL FRICTION OF FLUIDS

general equations have the same form whether the fluid be referred
to the fixed or moveable axes. But on the latter supposition
the equations of condition (30) become rigorously exact. Hence
equation (40) gives correctly the solution of the problem, inde¬
pendently of the restriction that the maximum excursion of the
sphere be small compared with its radius, provided we suppose
the polar co-ordinates r , 8 measured from the centre of the sphere
in its actual, not its mean position. Similar remarks apply to the
problem of the cylinder. Moreover, in the case of a sphere oscil¬
lating within a concentric spherical envelope, it is not necessary,
in order to employ the solution obtained in Section II., that the
maximum excursion of the sphere be small compared with its
radius; it is sufficient that it be small compared with the radius
of the envelope.

These are points of great importance, because the excursions of
an oscillating sphere in a pendulum experiment are not by any
means extremely small compared with the radius of the sphere ;
and in the case of a narrow cylinder, such as the suspending wire,
so far from the maximum excursion being small compared with
the radius of the cylinder, it is, on the contrary, the radius which
is small compared with the maximum excursion.

42. Let us now return to the case of the uniform motion of a
sphere. In order to obtain directly the expression for the resist¬
ance of the fluid, it would be requisite first to find p , then to get
P r and Tq from (46), or at least to get the values of these functions
for r~ a, and lastly to substitute in (47) and perform the integra¬
tion. We should obtain p by integrating the expression for dp got
from (16) and (17). It would be requisite first to express u and q
in terms of then to transform the expression for dp so as to
involve polar co-ordinates, and then substitute for ^ its value given
by (121); or else to express the right-hand member of (121) by the
co-ordinates so, us, and substitute in the expression for dp *. We

* The equations (16), (17) give, after a troublesome transformation to polar co¬
ordinates,

dp __ p d f d 2 sin d d l d p d\

dr “~r 2 sin# dQ \dr 2 * r 2 dd sin# dd p dt) ^ .

dp_ p d f d 2 sin# d 1 d p d\ , .

dd~~ sin# dr \dr 2 + r 2 dd sin# dd p dt) ^.

The expression for dp got from these equations is an exact differential by virtue
of the equation which determines \j /; and in the problems considered in Section II.

ON THE MOTION OF PENDULUMS.

59

have seen, however, that the results applicable to uniform motion
may be deduced as limiting cases of those which relate to oscil¬
latory motion, and consequently, we may make use of the ex¬
pression for F already worked out. Writing V for ce^~ lnt in the
first equation of Art. 20, expressing m in terms of n , and then
making n vanish, we get

— .F= 67rfi pa V. .(126),

and - F is the resistance required.

This equation may be employed to determine the terminal
velocity of a sphere ascending or descending in a fluid, provided
the motion he so slow that the square of the velocity may he
neglected. It has been shewn experimentally by Coulomb *, that
in the case of very slow motions, the resistance of a fluid depends
partly on the square and partly on the first power of the velocity.
The formula (126) determines, in the particular case of a sphere,
that part of the whole resistance which depends on the first power
of the velocity, even though the part which depends on the square
of the velocity be not wholly insensible.

It is particularly to be remarked, that according to the formula
(126), the resistance varies not as the surface but as the radius of
the sphere, and consequently the quotient of the resistance divided
by the mass increases in a higher ratio, as the radius diminishes,
than if the resistance varied as the surface. Accordingly, fine
powders remain nearly suspended in a fluid of widely different
specific gravity.

and in the present Section \£> has the form Sk sin 2 0, where Sk is independent of 0.
Hence we get from (&), by integrating partially with respect to 0,

p — p cos0

£

dr

Ur 2

2

f 5

p*\

/jl dt)

*

(c).

It is unnecessary to add an arbitrary function of r, because if X (?•) be such
a function which we suppose added to the right-hand member of (c), we must
determine X by substituting in (a). The resulting expression for X' (?•) cannot con¬
tain 0, inasmuch as the expression for dp is an exact differential, but it is composed
of terms which all involve cos0 as a factor, and therefore we know, without working
out, that these terms must destroy one another. Hence X (r) must be constant, or
at most be a function of t , which we may suppose included in n. X (r) will in fact
be equal to zero if n be the equilibrium pressure at the depth at which fgdz '
vanishes.

* M6moires de VInstitute Torn* in. p. 246.

60 ON THE EFFECT OF THE INTERNAL FRICTION OF FLUIDS

43. When the motion is so slow that the part of the re¬
sistance which depends on the square of the velocity may be
neglected, we have, supposing V to be the terminal velocity,
— _F= irg (a — p) a\ where g is the force of gravity, and <r, which
is supposed greater than p , the density of the sphere. Substituting
in (126) we get

. (m) '

Let us apply this equation to determine the terminal velocity
of a globule of water forming part of a cloud. Putting g= 386,
/z/ = (*116) 2 , an inch being the unit of length, and supposing
op" 1 — 1 = 1000, in order to allow a little for the rarity of the air
at the height of the cloud, we get V = 6372 x 1000a 2 . Thus, for a
globule the one thousandth of an inch in diameter, we have
V = 1*593 inch per second. For a globule the one ten thousandth
of an inch in diameter, the terminal velocity would be a hundred
times smaller, so as not to amount to the one sixtieth part of an
inch per second.

We may form a very good judgment of the magnitude of that
part of the resistance which varies as the square of the velocity,
and which is the only kind of resistance that could exist if the
pressure were equal in all directions, by calculating the numerical
value of the resistance according to the common theory, imperfect
though it be. It follows from this theory that if h be the height
due to the velocity V, the resistance is to the weight as 3 ph to 8 era.
For V= 1*593 inch per second, the resistance is not quite the one
four hundredth part of the weight; and for a sphere only the one
ten thousandth of an inch in diameter, moving with the velocity
calculated from the formula (127), the ratio of the resistance to the
weight would be ten times as small. The terminal velocities of
the globules calculated from the common theory would be 32*07
and 10*14 inches per second, instead of only 1*593 and *01593 inch.
It appears then that the apparent suspension of the clouds is
mainly due to the internal friction of air.

44. The resistance to the globule has here been determined
as if the globule were a solid sphere. In strictness, account ought
to be taken of the relative motion of the fluid particles forming the
globule itself. Although it may readily be imagined that no

ON THE MOTION OF PENDULUMS.

material change would thus be made in the numerical result, *
may be worth while to point out the mode of solution of the
problem. Suppose the globule preserved in a strictly spherical
shape by capillary attraction, which will very nearly indeed be the
case. Conceive a velocity equal and opposite to that of the
globule impressed both on the globule and on the surrounding
fluid, which will reduce the problem to one of steady motion.
Let fa, &c. refer to the fluid forming the globule, and assume
^ - f t (r) sin 2 0. Then we get on changing the constants in (119)
f t (r) = A^ 1 + B x r + G x r 2 + D X P.

The arbitrary constants, A x) B x vanish by the condition that the
velocity shall not become infinite at the centre. There remain
the two arbitrary constants G v D x to be determined, in addition to
those which appeared in the former problem. But we have now
four instead of two equations of condition which have to be satis¬
fied at the surface of the sphere, which are that

R=0, R x = 0 } © = © x , T d = T 10) when r = a .(128).

We shall thus have the same number of arbitrary constants as condi¬
tions to be satisfied. Now T l0 will involve /q as a coefficient, just as
T e involves pip or g,; and /q, which refers to water, is much larger
than /-&, which refers to air, although p! is larger than p, x . Hence
the results will be nearly the same as if we had taken pu x = oo , or
regarded the sphere as solid.

If, however, instead of a globule of liquid descending in a gas
we have a very small bubble ascending in a liquid, we must not
treat the bubble as a solid sphere. We may in this case also
neglect the motion of the fluid forming the sphere, but we have
now arrived at the other extreme case of the general problem, and
the two equations of condition which have to be satisfied at the
surface of the sphere are that R = 0 and T e = 0 when r = a, instead
of R = 0 and © = 0, when r = a.

The equation of condition T e = 0 which applies to a bubble, as
well as the fourth of equations (128), will not be the true equa¬
tions, if forces arising from internal friction exist in the superficial
film of a fluid which are of a different order of magnitude from
those which exist throughout the mass. At the end of the memoir
already referred to, Coulomb states that in very slow motions the
resistance of bodies not completely immersed in a liquid is much
greater than that of bodies wholly immersed, and promises to

62 ON THE EFFECT OF THE INTERNAL FRICTION OF FLUIDS

communicate a second memoir in continuation of the first. This
memoir, so far as I can find out, has never appeared. Should the
existence of such forces in the superficial film of a liquid be made
out, the results deduced from the theory of internal friction will
be modified in a manner analogous to that in which the results
deduced from the common principles of hydrostatics are modified
by capillary attraction. It may be remarked that we have nothing
to do with forces of this kind in considering the motion of pendu¬
lums in air, or even in considering the oscillations of a sphere in
water, except as regards the very minute fraction of the whole
effect which relates to the resistance experienced by the suspending
wire in the immediate neighbourhood of the free surface.

It may readily be seen that the effect of a set of forces in the
superficial film of a liquid offering a peculiar resistance to the
relative motion of the particles would be, to make the resistance of
a gas to a descending globule agree still more closely with the
result obtained by regarding the globule as solid, while the re¬
sistance experienced by an ascending bubble would be materially
increased, and made to approach to that which would belong to a
solid sphere of the same size without mass, or more strictly, with a
mass only equal to that of the gas forming the bubble. Possibly
the determination of the velocity of ascent of very small bubbles
may turn out to be a good mode of measuring the amount of fric¬
tion in the superficial film of a liquid, if it be true that forces of
this kind have any existence. But any investigation relating to
such a subject would at present be premature.

45 . Let us now attempt to determine the uniform motion of a
fluid about an infinite cylinder. Employing the notation of Section
III., and reducing the problem to one of steady motion as in Art.
39 , we obtain the same equations of condition (116), (117), as in
the case of the sphere. Assuming % = sin 6 F (r), and substituting
in the equation obtained from (69) by transforming to polar co¬
ordinates and leaving out the terms which involve d/dt, we get

.

The integral of this equation may readily be obtained by mul¬
tiplying the last term of the operating factor by (1 -b S) 2 , integrating
the transformed equation, and then making 8 vanish. It is

F (r) = Ar~ l Br+ Or log r + Dr* .(130)

ON THE MOTION OF PENDULUMS.

63

which gives

R = = ( Ar~ 2 + B + <71og r + Dr*) cos Q

rao

© = - & = (Ar* - B - C - G log r - 32V) sin 6.

CLF

The first of the equations of condition (117) requires that
0=0, D=0, B = -V,

which also satisfies the second. We have thus only one arbitrary
constant left whereby to satisfy the two equations of condition
(116), and the same value of A will not satisfy these two
equations.

46. It appears then that the supposition of steady motion is
inadmissible. It will be remembered that, in the case of the
sphere, the solution of the problem was only possible because it so
happened that the values of two arbitrary constants determined by
satisfying the first of the equations of condition (117) satisfied also
the second, which indicates that the solution was to a certain extent
tentative. We have evidently a right to conceive a sphere or infinite
cylinder to exist at rest in an infinite mass of fluid also at rest, to
suppose the sphere or cylinder to be then moved with a uniform
velocity V\ and to propose for determination the motion of the fluid
at the end of the time t. But we have no right to assume that the
motion approaches a permanent state as t increases indefinitely.
We may follow either of two courses. We may proceed to solve
the general problem in which the sphere or cylinder is supposed
to move from rest, and then examine what results we obtain by
supposing t to increase indefinitely, or else we may assume for
trial that the motion is steady, and proceed to inquire whether wc
can satisfy all the conditions of the problem on this supposition.
The former course would have the disadvantage of requiring a
complicated analysis for the sake of obtaining a comparatively
simple result, and it is even possible that the solution of the
problem might baffle us altogether; but if we adopt the latter
course, we must not forget that the equations with which we work
are only provisional.

It might be objected that the impossibility of satisfying the
conditions of the problem on the hypothesis of steady motion

64 ON THE EFFECT OF THE INTERNAL FRICTION OF FLUIDS

arose from our assumption that sin 0 was a factor of the other
factor being independent of 6. This however is not the case.
For, for given values of r and t, % is a finite function of 6 from
0=Oto0 = 7 r. We have a right to suppose ^ to vanish at any
point of the axis of x positive that we please; and if we suppose
X to vanish at one such point, it may be shewn as in the note
to Art. 15, that % will vanish at all points of the axis of x positive
or x negative. Hence % may be expanded in a convergent series
of sines of 0 and its multiples; and since % and its derivatives with
respect to 0 alter continuously with 0 , the expansions of the
derivatives will be got by direct differentiation*. This being
true for all other pairs of values of r and t , % can in general
be expanded in a convergent series of sines of 0 and its multiples;
but the coefficients, instead of being constant, will be functions of
r and t, or in the particular case of steady motion, functions of r
alone. Now a very slight examination of the general equations
will suffice to shew that the coefficients of the sines of the different
multiples of 0 remain perfectly independent throughout the whole
process, and consequently had we employed the general expansion,
we should have been led to the very same conclusions which have
been deduced from the assumed form of %

47. If we take the impossibility of the existence of a limiting
state of motion, which has just been established, in connexion with
the results obtained in Section III., we shall be able to understand
the general nature of the motion of the fluid around an infinite
cylinder which is at first at rest, and is then moved on indefinitely
with a uniform velocity.

The fluid being treated as incompressible, the first motion
which takes place is impulsive. Since the terms depending on
the internal friction will not appear in the calculation of this
motion, we may employ the ordinary equations of hydrodynamics.
The result, which is easily obtained, is

Yd 2

Rdr + ®rd0 = d</>, where </> =-— cos .(131).

* See a paper “On the Critical Values of the Sums of Periodic Series,” Camb.
Phil. Trans. Vol. viii. p. 533. [Ante, Vol. i. p. 236.]

+ According to these equations, the fluid flows past the surface of the cylinder
with a finite velocity. At the end of the small time t' after the impact, the friction
has reduced the velocity of the fluid in contact with the cylinder to that of the

ON THE MOTION OF PENDULUMS.

65

As the cylinder moves on, it carries more and more of the fluid
with it, in consequence of friction. For the sake of precision, let
the quantity carried by the element dl of the cylinder be defined
to be that which, moving with the velocity V, would have the
same momentum in the direction of the motion that is actually
possessed by the elementary portion of fluid which is contained
between two parallel infinite planes drawn perpendicular to the
axis of the cylinder, at an interval dl, the particles composing
which are moving with velocities that vary from V to zero in
passing from the surface outwards. The pressure of the cylinder
on the fluid continually tends to increase the quantity of fluid
which it carries with it, while the friction of the fluid at a distance
from the cylinder continually tends to diminish it. In the case of
a sphere, these two causes eventually counteract each other, and
the motion becomes uniform. But in the case of a cylinder, the
increase in the quantity of fluid carried continually gains on the
decrease due to the friction of the surrounding fluid, and the
quantity carried increases indefinitely as the cylinder moves on.
The rate at which the quantity carried is increased decreases con¬
tinually, because the motion of the fluid in the neighbourhood of
the cylinder becomes more and more nearly a simple motion of
translation equal to that of the cylinder itself, and therefore the
rate at which the quantity of fluid carried is increased would
become smaller and smaller, e\ren were no resistance offered by
the surrounding fluid.

The correctness of this explanation is confirmed by the follow¬
ing considerations. Suppose that F(r) had been given by the
equation

F(r) = Ar' 1 4 - Br -f CV 1-5 -f Dr*

instead of (130), S being a small positive quantity. On this sup¬
position it would have been possible to satisfy all the equations of

cylinder itself, and the tangential velocity alters very rapidly in passing from the
surface outwards. At a small distance s from the surface of the cylindor, the rela¬
tive velocity of the fluid and the cylinder, in a direction tangential to the surface,
is a function of the independent variables £', k, which vanishes with a for any given
value of however small, but which for any given value of **, however small,
approaches indefinitely to the quantity determined by (131) as t vanishes. The
communication of lateral motion is similar to the communication of temperature
when the surface of a body has its temperature instantaneously raised or lowered
by a finite quantity.

S. III.

5

66 ON THE EFFECT OF THE INTERNAL FRICTION OF FLUIDS

condition, and therefore steady motion would have been possible.
By determining the arbitrary constants, and substituting in % we
should have obtained

, Tr , Bar 2 fr

f = ^ "5- --- +

1“<

B=V\ -

r a 2 — S

fa\

2 — S \r/ 2 — 8 \a

sin #,
cos #,

©=F

2 — S

\rj

2 . 2(l-$)/r\- s ) • /,

+l —krrU)

Since B is supposed to be extremely small, it follows from these
expressions that when r is not greater than a moderate multiple
of a , the velocities JR, © are extremely small; but, however small
be S, we have only to go far enough from the cylinder in order to
find velocities as nearly equal to — V cos#, 4* Fsin# as we please.
But the distance from the cylinder to which we must proceed in
order to find velocities JR, © which do not differ from their limiting
values — Fcos#, +V sin# by more than certain given quantities,
increases indefinitely as S decreases. Hence, restoring to the fluid
and the cylinder the velocity F, we see that in the neighbourhood
of the cylinder the motion of the fluid does not sensibly differ from
a motion of translation, the same as that of the cylinder itself,
while the distance to which the cylinder exerts a sensible influence
in disturbing the motion of the fluid increases indefinitely as
8 decreases.

48. When we have formed the equations of motion of a fluid
on any particular dynamical hypothesis, it becomes a perfectly
definite mathematical problem to determine the motion of the
fluid when a given solid, initially at rest as well as the fluid, is
moved in a given manner, or to discuss the character of the ana¬
lytical solution in any extreme case proposed. It is quite another
thing to enquire how far the principles which furnished the mathe¬
matical data of the problem may be modified in extreme cases,
or what will be the nature of the actual motion in such cases.
Let us regard in this point of view the case considered in the
preceding article as a mathematical problem. When the quantity
of fluid carried with the cylinder becomes considerable compared
with the quantity displaced, it would seem that the motion must
become unstable, in the sense in which the motion of a sphere

ON THE MOTION OF PENDULUMS.

67

rolling down the highest generating line of an inclined cylinder
may be said to be unstable. But besides the instability, it may
not be safe in such an extreme case to neglect the terms depending
on the square of the velocity, not that they become unusually
large in themselves, but only unusually large compared with the
terms retained, because when the relative motions of neighbouring
portions of the fluid become very small, the tangential pressures
which arise from friction become very small likewise.

Now the general character of the motion must be nearly the
same whether the velocity of the cylinder be constant, or vary
slowly with the time, so that it does not vary materially when the
cylinder passes through a space equal to a small multiple of its
radius. To return to the problem considered in Section III., it
would seem that when the radius of the cylinder is very small, the
motion which would be expressed by the formulas of that Section
would be unstable. This might very well be the case with the
fine wires used in supporting the spheres employed in pendulum
experiments. If so, the quantity of fluid carried by the wire
would be diminished, portions being continually left behind and
forming eddies. The resistance to the wire would on the whole be
increased, and would moreover approximate to a resistance which
would be a function of the velocity. Hence, so far as depends on
the wire, the arc of oscillation would be more affected by the
resistance of the air than would follow from the formulas of
Section III. Whether the effect on the time of oscillation would
be greater or less than that expressed by the formulcc is difficult
to say, because the increase of resistance would tend to increase
the effect on the time of vibration, while on the other hand the
approximation of the law of resistance to that of a function of the
velocity would tend to diminish it.

Section V.

On the effect of internal friction in causing the motion, of a fluid
to subside . Application to the case of os dilatorg waves.

49. We have already had instances of the effect of friction in
causing a gradual subsidence in the motion of a solid oscillating in
a fluid; but a result may easily be obtained from the equations of

5—2

68 ON THE EFFECT OF THE INTERNAL FRICTION OF FLUIDS

motion in their most general shape, which shews very clearly the
effect of friction in continually consuming a portion of the work
of the forces acting on the fluid.

Let P t , P 2 , P 3 be the three normal, and T 19 P 2 , T 3 the three
tangential pressures in the direction of three rectangular planes
parallel to the co-ordinate planes, and let D be the symbol of
differentiation with respect to t when the particle and not the
point of space remains the same. Then the general equations
applicable to a heterogeneous fluid (the equations (10) of my
former paper) are

(DU y

P{m- X

dT^ dT 2
dx dy dz

,.(132),

with the two other equations which may be written down from
symmetry. The pressures P 1? &c. are given by the equations

P = •

(dv dw\

and four other similar equations. In these equations

38 =

du dv dw
dx dy dz

.(134).

At the end of the time t let V be the vis viva of a limited
portion of the fluid, occupying the space which lies inside the
closed surface S, and let V+DV be the vis viva of the same mass
at the end of the time t + Dt. Then

V=JJJp (' u 2 + v 2 4- w 2 ) dx dy dz,

BV= Wt flf p i u ik +v m +w m) (lx dy dz . (135)>

the triple integrals extending throughout the space bounded by S.
])u

Substituting now for , &c. their values given by the equations
of the system (132), we get

DV = 2 Dt fffp (uX + vY + wZ) dx dy dz

d l\ + d ^ + ^\

dy dz djX)

( ~f‘ 8 + ~r~ 2 + ) \ dxdy dz

\dz dx dy J) J

- 2DtJJJ |w (£■ + !|> + + r r+

+ W - 4 - 4 - — 1 ^

(136).

ON THE MOTION OF PENDULUMS.

69

The first part of this expression is evidently twice the work,
during the time Dt, of the external forces which act all over the
mass. The second part becomes after integration by parts

- 2Dt Jf (uP 1 + vT 3 + wT^) dy dz — 2DtJf(vP 2 + wT x + uT s ) dz dx

— 2Dt fj ( wP 3 + uT 2 + vT x ) dx dy

+ 2Dt

Iff

du -p. dv „ dw 7 -v , (dv dw\ m fdw
Tx P ^dy P ^d^ P ^{Tz + dy) T * + f

dx

+ (f + sW ***•

<fo\ _
dz) 2

The double integrals in this expression are to be extended over
the whole surface S. If dS be an element of this surface, l\ m\ n'
the direction-cosines of the normal drawn outwards at dS , we may-
write I'dS, m'dS , n'dS for dydz , dzdx, dxdy. The second part of
DV thus becomes

- 2 DtJJ (l'P 1 + m'T z + n%) + v (m'P 2 + riT x + VT Z )

+ w(n'P 3 +l'T 2 + m'T 1 )}dS.

The coefficients of u } v , w in this expression are the resolved parts,
in the direction of x, y, z, of the pressure on a plane in the direc¬
tion of the elementary surface dS, whence it appears that the
expression itself denotes twice the work of the pressures applied
to the surface of the portion of fluid that we are considering.

On substituting for P v &c. their values given by the equations
(133), (134), we get for the last part of DV

fff fdu dv dw\ 7 , 1

♦“‘jJJnE+s+s) *

\ \dx,

dz)

2 /du dv dw
^ \dx dy "dz

dv did
dz dy.

In this expression p denotes, in the case of an elastic fluid, the
pressure statically corresponding to the density which actually
exists about the point whose co-ordinates are x, y } z , and the part
of the expression which contains p denotes twice the work con¬
verted into vis viva in consequence of internal expansions, and
arising from the forces on which the elasticity depends. The last
part of the expression is essentially negative, or at least cannot be

70 ON THE EFFECT OF THE INTERNAL FRICTION OF FLUIDS

positive, and can only vanish in one very particular case. It
denotes the vis viva consumed, or twice the work lost in the
system during the time dt, in consequence of internal friction.
According to the very important theory of Mr Joule, which is
founded on a set of most striking and satisfactory experiments,
the work thus apparently lost is in fact converted into heat, at
such a rate, that the work expressed by the descent of 772 lbs
through one foot, supplies the quantity of heat required to raise
1 lb. of water through 1° of Fahrenheit’s thermometer.

50. The triple integral containing fx can only vanish when
the differential coefficients of u, v y w satisfy the five following
equations,

du _dv __ dw
dx dy~~ dz 3
dv dw __ . dw du

dz dy 3 dx dz 3

\

du dv
dy^ dx

...(137).

These equations give immediately the following expressions
for the differentials of u, v, w, in which the co-ordinates alone are
supposed to vary, the time being constant:

du = Mx — (o"'dy + a /'dz \

dv = My — co'dz + oa r "dx l.(138).

dw — Mz — co"dx + co'dy j

In these equations S, co', co", co'" are certain functions of which the
forms are defined by the equations (138), but need not at present
be considered. It follows from equations (138) that the motion of
each element of the fluid within the surface S is compounded of
a motion of translation, a motion of rotation, and a motion of
dilatation alike in all directions. So far as regards the first two
kinds of motion, the fluid element moves like a solid, and of
course there is nothing to call internal friction into play. For
the reasons stated in my former paper, I was led to assume that a
motion of dilatation alike in all directions (which of course can
only exist in the case of an elastic fluid) has no effect in causing
the pressure to differ from the statical pressure corresponding to
the actual density, that is, in occasioning a violation of the
functional relation commonly supposed to exist between the
pressure, density, and temperature. The reader will observe that

ON THE MOTION OF PENDULUMS.

71

this is a totally different thing from assuming that a motion of
dilatation has no effect on the pressure at all.

"When the fluid is incompressible 8 = 0, and it may be proved
without difficulty that a/, a>", a/" are constant, that is to say,
constant so far as the co-ordinates are concerned. In this case we
get by integrating equations (137)

u — a — a)'"y + u! f z )

v ~ b — cd z + a'"ail .(139).

w~c — o/'x + co'y j

Hence, in the case of an incompressible fluid, unless the whole
mass comprised within the surface S move together like a solid,
there cannot fail to be a certain portion of vis viva lost by internal
friction. In the case of an elastic fluid, the motion which may
take place without causing a loss of vis viva in consequence of
friction is somewhat more general, and corresponds to velocities
u + Au , v + Av, w -f Aw, where u, v , w are the same as in (139),
and

Au = 8 ^ + 2 (<xx + f3y + 7 z) x — ol(x 2 + y 2 -f /),

with similar expressions for Aw and Aw. In these expressions
a, /3, 7 are three constants symmetrically related to x, y, z , and
8 is a constant which has the same relation to each of the
co-ordinates*.

51. By means of the expression given in Art. 49, for the loss
of vis viva due to internal friction, we may readily obtain a very
approximate solution of the problem : To determine the rate at
which the motion subsides, in consequence of internal friction, in
the case of a series of oscillatory waves propagated along the
surface of a liquid.

Let the vertical plane of xy be parallel to the plane of motion,
and let y be measured vertically downwards from the mean
surface; and for simplicity’s sake suppose the depth of the fluid
very great compared with the length of a wave, and the motion so
small that the square of the velocity may be neglected. In the
case of motion which we are considering, udx + vdy is an exact
differential dcfr when friction is neglected, and

<£ = ce~ my sin(m^“- nt) .(140),

* See note C. at the end.

72 ON THE EFFECT OF THE INTERNAL FRICTION OF FLUIDS

where c, m, n are three constants, of which the last two are con¬
nected by a relation which it is not necessary to write down.
We may continue to employ this equation as a near approximation
when friction is taken into account, provided we suppose c, instead
of being constant, to be a parameter which varies slowly with the
time. Let V be the vis viva of a given portion of the fluid at the
end of the time t, then

F= pcWS!fe~ 2my dxdydz .(141).

But by means of the expression given in Art. 49, we get for the
ioss of vis viva* during the time dt, observing that in the present
case p is constant, w = 0, § = 0, and udx + vdy = d<fi, where <j£> is
independent of z,

which becomes, on substituting for its value,

8^tc 2 m 4 dtfjje~ 2mu dx dy dz.

But we get from (141) for the decrement of vis viva of the same
mass arising from the variation of the parameter c

— 2 pm 2 c — dtfjj€’ 2my dx dy dz.

Equating the two expressions for the decrement of vis viva ,
putting for m its value 27 tX~ 1 , where X is the length of a wave,
replacing p by pup, integrating, and supposing c 0 to be the initial
value of c, we get

IGrr^fx't
c = C 0 € .

It will presently appear that the value of p for water is
about 0*0564, an inch and a second being the units of space and

* [There is an oversight here, which M. Boussinescq has pointed out (Metnoires
des Savans Strangers , Tome xxiv. No. 2, p. 34). I should have said “the loss of
energy .” Now in a series of waves of small disturbance the total energy is half
kinetic and half potential. Hence the reduction of energy consequent upon a reduc¬
tion in the amplitude falls half on the kinetic and half on the potential energy.
Hence the reduction of the kmetic energy or vis viva is only half of that given by
the formula in the text, and therefore the expression for dcjdt is twice what it ought
to be. Hence the numerical coefficient in the index of the exponential should be
8 instead of 16; and retaining the same numerical data as in the examples, we
should have for the ripples c : c 0 :: 1 : 0-5337, and the height of the long waves
would be reduced in a day by little more than the one four-hundredth part.]

ON THE MOTION OF PENDULUMS.

73

time. Suppose first that \ is two inches, and t ten seconds.
Then = 1*256, and c:c 0 ::l : 0*2848, so that the height

of the waves, which varies as c, is only about a quarter of what it
was. Accordingly, the ripples excited on a small pool by a puff
of wind rapidly subside when the exciting cause ceases to act.

Now suppose that X is 40 fathoms or 2880 inches, and that
t is 86400 seconds or a whole day. In this case 167 t 2 //£V 2 is equal
to only 0*005232, so that by the end of an entire day, in which
time waves of this length would travel 574 English miles, the height
would be diminished by little more than the one two-hundredth
part in consequence of friction. Accordingly, the long swells of
the ocean are but little allayed by friction, and at last break on
some shore situated at the distance of perhaps hundreds of miles
from the region where they were first excited.

52. It is worthy of remark, that in the case of a homogeneous
incompressible fluid, whenever udx -f vdy + wdz is an exact dif¬
ferential, not only are the ordinary equations of fluid motion
satisfied*, but the equations obtained when friction is taken into
account are satisfied likewise. It is only the equations of con¬
dition which belong to the boundaries of the fluid that are violated.
Hence any kind of motion which is possible according to the
ordinary equations, and which is such that udx + vdy + wdz is an
exact differential, is possible likewise when friction is taken into
account, provided we suppose a certain system of normal and
tangential pressures to act at the boundaries of the fluid, so as
to satisfy the equations of condition. The requisite system of
pressures is given by the system of equations (133). Since fju
disappears from the general equations (1), it follows that p is the
same function as before. Bat in the first case the system of
pressures at the surface was P 1 ~P 2 = P 3 —p ) 2\ = ]\ = T ? — 0.
Hence if A P 1 &c. be the additional pressures arising from friction,
we get from (133), observing that 3=0, and that udx + vdy + wdz
is an exact differential d<f >,

AP 1 =

- 2/j, -A

d‘4,

daf

A P„

AP 2 = — 2/i

2 /a,

d‘<f>

dz*

<P<£

(142),

* It is here supposed that the forces X, Y, Z are such that Xdx+ Ydy + Zdz is
an exact differential.

74 ON THE EFFECT OF THE INTERNAL FRICTION OF FLUIDS

Let dS be an element of the bounding surface, V, m, ri the
direction-cosines of the normal drawn outwards, AP, A Q, A R the
components in the direction of x, y , z of the additional pressure on
a plane in the direction of dS. Then by the formulae (9) of my
former paper applied to the equations (142), (143) we get

da? dx dy dx dz

...(144),

with similar expressions for A Q and AJS, and AP, AQ, Ai2
are the components of the pressure which must be applied at the
surface, in order to preserve the original motion unaltered by
friction.

53. Let us apply this method to the case of oscillatory waves,
considered in Art. 51. In this case the bounding surface is nearly
horizontal, and its vertical ordinates are very small, and since the
squares of small quantities are neglected, we may suppose the
surface to coincide with the plane of xz in calculating the system
of pressures which must be supplied, in order to keep up the
motion. Moreover, since the motion is symmetrical with respect
to the plane of xy , there will be no tangential pressure in the
direction of z, so that the only pressures we have to calculate are
AP 2 and AT s . We get from (140), (142), and (143), putting
y = 0 after differentiation,

AP 2 = — 2 fjbw?c sin (mx — nt), A T 8 = 2 y,m?c cos (mx — nt ).. .(145).

If u 1} Vj be the velocities of the surface, we get from (140),
putting y = 0 after differentiation,

u x = me cos (mx — nt), v x = — mo sin (mx — nt).. .(146).

It appears from (145) and (146) that the oblique pressure
which must be supplied at the surface in order to keep up the
motion is constant in magnitude, and always acts in the direction
in which the particles are moving.

ON THE MOTION OF PENDULUMS.

75

The work of this pressure during the time dt corresponding to
the element of surface dx dz, is equal to

dx dz(AT 3 ,u x dt + AP 2 . v 2 dt).

Hence the work exerted over a given portion of the surface is
equal to

2/j,m*c 2 dt Jfdx dz.

In the absence of pressures AP 2 , A T s at the surface, this work
must be supplied at the expense of vis viva . Hence 4yamVcfe ffdx dz
is the vis viva lost by friction, which agrees with the expression
obtained in Art. 51, as will be seen on performing in the latter the
integration with respect to y, the limits being y = 0 to y = 00 .

76 ON THE EFFECT OF THE INTERNAL FRICTION OF FLUIDS

PART II.

COMPARISON OF THEORY AND EXPERIMENT.
Section I.

Discussion of the Experiments of Daily , Bessel , Coulomb , and

Dubuat*.

54 The experiments discussed in this Section will be taken
in the order which is most convenient for discussion, which
happens to be almost exactly the reverse of the chronological
order. I commence with the experiments of the late Mr Baily,
which are described in the Philosophical Transactions for 1832, in
a memoir entitled “On the Correction of a Pendulum for the
Reduction to a Yacuum: together with Remarks on some anomalies
observed in Pendulum experiments.”

* [At the time when this paper was read, the relation between /ul and p cannot be
said to have been known. It is true that it may be inferred (at least for air, and
thence presumably for other gases) from certain of Graham’s experiments on the
transpiration of gases. These however had been but recently published, having
appeared in the Philosophical Transactions for 1846; and it was not till many years
afterwards, about 1859, that Maxwell first inferred from the kinetic theory of gases
the law that bears his name, namely that the coefficient of viscosity ju. is inde¬
pendent of the density.

In the comparison of theory and experiment as regards the effect of the presence
of air on the motion of pendulums, I relied mainly on the experiments of Baily,
which were made by a direct method, while at the same time they were conducted
with all the accuracy of modern physical research, and embraced a great variety of
forms of pendulum, many of them such as to admit of comparison with theory.

These experiments were strictly differential, giving the difference between the
time of vibration at atmospheric pressure and in rarefied air. Had the vacuum
been absolutely perfect, the difference would have given at once the effect of air at
the atmospheric pressure. Had it merely been very high, the effect of the residual
air on the time of vibration would have been insensible, and the result as regards
the time would still have been the same. It is true that the ichole effect of the
rarefied air would not thus disappear; as a result of Maxwell’s law it would tend, as
the exhaustion proceeded, to fall wholly on the arc of vibration, and to approach a
finite limit; and this limit would not begin to break down till an exhaustion was
reached comparable with the highest we have to deal with in radiometer vacua.

But in Baily’s experiments no high exhaustions were aimed at; the air was
merely pumped out till the pressure was reduced to about one inch of mercury, and

ON THE MOTION OF PENDULUMS.

77

The object of these experiments was, to determine by actual
observation the correction to the time of vibration due to the
presence of the air in the case of a great number of pendulums of
various forms. This was effected by placing each pendulum in
succession in a vacuum apparatus, by which means the pendulum,
without being dismounted, could be swung alternately under the
full atmospheric pressure, and in air so highly rarefied as nearly to
approach to a vacuum. The paper, as originally presented to the
Royal Society, contained the results obtained with 41 pendulums,
the same body being counted as a different pendulum when swung
in a different manner. Out of these, 14 are of such forms as
to admit of comparison with theory. An addition to the paper
contains the results obtained with 45 pendulums more, of which
24 admit of comparison with theory. The details of these
additional experiments are omitted, the results only being
given.

Baily has exhibited the results obtained with the several
pendulums in each of two ways, first, by the value of the factor n
by which the correction for buoyancy must be multiplied in order
to amount to the whole effect of the air as given by observation,
and, secondly, by the weight of air which must be conceived to be

the effect of the air was supposed to be arrived at by increasing the observed
difference in the times of vibration in the ratio of the difference of densities to the
atmospheric density. As the effect of the air at the lower density was too large to
he neglected, it was necessary, in order to compare with sufficient accuracy the
results of experiment with the formulae of this paper, to know the relation between
fx and p. As already mentioned, I assumed in accordance with what appeared to be
indicated by a single experiment of Sabine’s that p. varies as p, or in other words
that p! is independent of the density. The results of the experiments when thus
reduced seemed to indicate a most remarkable accord with theory.

When it became known that the law of nature is that p and not p/ is independent
of p, it seemed very strange that the experiments when reduced on the assump¬
tion of a wrong law as to the relation of p, to p should have led to such a
remarkable agreement with theory. I contemplated at one time undertaking the
re-computation of the whole series of Baily’s experiments here discussed in ac¬
cordance with Maxwell’s law, and it was this that delayed the reprinting of the
present paper. The value of the result at the present time would however hardly
repay the labour of the calculation, more especially as the remarkable agreement
between theory and observation notwithstanding the employment of a wrong law
as to the relation between p, and p admits of being readily explained, and the value
of p obtained as in the text of being very approximately corrected, in a very simple
manner. As however this would be too long for a footnote, I must reserve it for an
addition to be made at the end of the paper.]

78 ON THE EFFECT OF THE INTERNAL FRICTION OF FLUIDS

attached to the centre of gyration of the pendulum, adding to its
inertia without adding to its weight, in order that the increased
inertia, combined with the buoyancy of the air, may account for
the whole effect observed. I shall uniformly write n for Baily’s
in order to distinguish it from the n of Part I. of the present
paper, which has a totally different meaning. In the case of a
pendulum oscillating in air, it will be sufficient, unless the
pendulum be composed of extremely light materials, to add
together the effects of buoyancy and inertia. Hence if the
pendulum consist of a sphere attached to a fine wire of which the
effect is neglected, or else of a uniform cylindrical rod, we may
suppose n = 1 + Jc, where k is the factor so denoted in Part I.; so
that if M be the mass of air displaced, kM will be the mass which
we must suppose collected at the centre of the sphere, or dis¬
tributed uniformly along the axis of the cylinder, in order to
express the effect of the inertia of the air. The second mode
of exhibiting the effect of the air was suggested by Mr Airy,
and is better adapted than the former for investigating the
effect of the several pieces of which a pendulum of complicated
form is composed. Since the value of the factor n and that of the
weight of air are merely two different expressions for the result of
the same experiment, it would be sufficient to compare either
with the result calculated from theory. In some cases, however, I
have computed both. In almost all the calculations 1 have em¬
ployed 4-figure logarithms. The experimental result is sometimes
exhibited to four figures, but no reliance can be placed on the
last. In fact, in the best observations, the mean error in different
determinations of n for the same pendulum appears to have been
about the one-hundredth part of the whole, and that it should be
so small, is a proof of the extreme care with which the experi¬
ments must have been performed.

55. I commence with the 13th set of experiments —Results
with plain cylindrical rods —page 441. This set contains three
pendulums, each consisting of a long rod attached to a knife-edge
apparatus. The result obtained with each pendulum furnishes an
equation for the determination of //, and the theory is to be tested
by the accordance or discordance of the values so obtained. The
principal steps of the calculation are contained in the following
table.

ON THE MOTION OF PENDULUMS.

79

Determination of \/ja by means of Baily’s experiments with
plain cylindrical rods .

Pendulum rod

No.

Diameter

2 a

Time of
vibration.

T

tt

by experi¬
ment

Copper, 58*8 inches long

21

0*410

1*0136

2*932

Brass, 56*4 .

43

0*185

0*9933

4*083

Steel, 56*4 .

44

0*072

0*9933

7*530

No.

Correction
for confined
space (by
theory)

Deduced
value of
k

by experi¬
ment

Corre¬
sponding
value of
m

Resulting
value of

Vm-'

21

-0*009

1*923

1*5445

0*1166

43

-0*002

3*081

*0*7000

0*1175

44

6*530

0*2822

0*1134

In this table the first column explains itself. The next
contains the reference number. In the case of the copper rod
I have replaced 42 by 21, under which number the details of
the experiment will be found. The diameters of the rods are
expressed in decimals of an inch. The time of vibration of the
pendulum No. 21 may be got from the tables at the end of
Baily’s memoir, which contain the details of the experiments.
Nos. 43 and 44 belong to the “ additional experiments,” of which
all the details are suppressed. Baily has not even given the
times of vibration, not having been aware of the circumstance,
indicated by the theory of this paper, that the factor n and
the weight of air which must be conceived as dragged by the
pendulum are functions of the time of vibration. Accordingly,
in the cases of the pendulums Nos. 43 and 44, and in all similar
cases, I have calculated the time of vibration by the ordinary
formulae of dynamics. In calculating t, I have added 1*55 inch,
the length of the shank of the knife-edge apparatus, to the length
of the rods. The result so obtained is abundantly accurate enough
for my purpose. Had the rod, retaining its actual length, been
supposed to begin directly at the knife-edge, the error thence

80 ON THE EFFECT OF THE INTERNAL FRICTION OF FLUIDS

resulting in the value of t, or rather the corresponding error
in the calculated value of n or k , might just have been sensible.
The fifth column in the above table is copied from Baily’s table.
The next contains a small correction necessary to reduce the value
of it got from observation to what would have been got from
observations made in an unlimited mass of fluid. It is calculated
from the formula 2a 2 ( b 2 — a 2 )" 1 or 2& 2 £f 2 nearly, which is obtained
from the ordinary equations of hydrodynamics, and therefore it
cannot be regarded as more than a rude approximation. It will
be useful, however, as affording an estimate of the magnitude of
the effect produced by confining the air. The diameter of the
vacuum tube (whether external or internal is not specified) is
stated to have been six inches and a half, whence 2 b — 6'5. The
values of k given in the next column are obtained by applying the
correction for confined space to Baily’s values of n, and subtracting
unity. The value of ill corresponding to each value of k was got
by interpolation from the table near the end of Section III. of the
former part of this paper. For k= 1*923 the interpolation is easy.
The value 3*081 happens to be almost exactly found in the table.
For k = 6*530, a remark already made will be found to be of
importance, namely, that the first differences of m 2 (&—1) are
nearly constant. The last column contains the value of \jfji
obtained from the equation

which contains the definition of m.

It will be observed that the three values of \J/ju are nearly
identical. Of course any theory professing to account for a set of
experiments by means of a particular value of a disposable con¬
stant, when applied to the experiments would lead to nearly the
same numerical value of the constant if the experiments were
made under nearly the same circumstances. But in the present
case the circumstances of the experiments are widely different.
The diameter of the steel rod is little more than the sixth part of
that of the copper rod, and the value of Jc obtained by experiment
for the steel rod is more than three times as great as that obtained
for the copper rod. It is a simple consequence of the ordinary
theory of hydrodynamics that in the case of a long rod oscillating

ON THE MOTION OF PENDULUMS.

81

in an unlimited fluid k = 1 , and we see that this value of k must
be multiplied, in round numbers, by 2, by 3, and by 6£, in order
to account for the observed effect. The value 1*5445 of HI is so
large that the descending series comes into play in the calculation
of the function k , while 0*2822 is so small that the ascending
series are rapidly convergent. Hence the near agreement of the
values of \jyi! deduced from the three experiments is a striking
confirmation of the theory. The mean of the three is 0*1158, but
of course the last figure cannot be trusted. I shall accordingly
assume as the value of the square root of the index of friction of
air in its average state of pressure, temperature, and moisture

VV = 0*116.

It is to be remembered that yV expresses a length divided by
the square root of a time, and that the numerical value above
given is adapted to an English inch as the unit of length, and a
second of mean solar time as the unit of time.

56. I now proceed to compare the observed values of tt with
those calculated from theory with the assumed value of I
begin with the same cylindrical rods as before, together with the
long brass tubes Nos. 35 to 38. The diameter of this tube was
1*5 inch, and its length 56 inches. The ends were open, but as
the included air was treated by Mr Baily in the reduction of his
experiments as if it formed part of the pendulum, we may regard
the pendulum as a solid rod. The tube was furnished with six
agate planes, represented in the wood-cut at page 417, which
rested on fixed knife-edges. The pendulums Nos. 35, 36, 37, and
38 consisted of the same tube swung on the planes marked A,C,a, c.
In air the pendulum swung at the rate of about 90080 vibrations
in a day, so that t = 0*9596 nearly. The values of n obtained
with the end planes A , c were slightly though sensibly greater
than the values obtained with the mean planes (7, a . I shall
suppose the mean of the four values of n, namely 2*290, to be the
result of the experiments. In the following table the difference
between the theoretical and experimental values of n is exhibited
both by decimals and as a fractional part of the former of these
values.

s. III.

6

82 ON THE EFFECT OF THE INTERNAL FRICTION OF FLUIDS

Bailys results with a long brass tube and with long cylindrical
rods .

No.

2 a.

t

m

!

k

Add for
confined
space.

i

Total n

by

theory.

!

n

by expe¬
riment.

l

Difference.

35 to 38

1*5

5*849

1-242

0-122

2-364

2-290 i

-0-074, or

21 or 42

0*410

1-555

1-917

, 0-009

2-926

2-932

+ 0"006, or +-jJt)

43

0*185

0-7089

3-055

0-002 I

4-057

4-083

+0'026, or + j4y

44

0-072

0-2759

l

6-670

7-670

7-530

-0-140, or -Ji

It will be seen at once how closely the experiments are
represented by theory. The largest proportionate difference
occurs in the case of the brass tube, and even that is less
than one-thirtieth. A glance at Baily’s wood-cut at page 417
will shew that the six planes with which the tube was furnished
caused the whole figure to deviate sensibly from the cylindrical
form. Moreover the resistance experienced by each element of
the cylinder has been calculated by supposing the element in
question to belong to an infinite cylinder oscillating with the
same linear velocity, and the resistance thus determined must
be a little too great in the immediate neighbourhood of the ends
of the cylinder, where the free motion of the air is less impeded
than it would be if the cylinder were prolonged. Lastly, the
correction for confined space is calculated according to the
ordinary equations of hydrodynamics, and on that account, as
well as on account of the abrupt termination of the cylinder,
will be only approximate. The small discrepancy between theory
and observation, as well as the small difference (amounting to
about the l-83rd of the whole) detected by experiment between the
results obtained with the extreme planes and those obtained with
the mean planes, may reasonably be attributed to some such
causes as those just mentioned. In the case of the steel rod
or wire, the difference between theory and observation may be
altogether removed by supposing a very small error to have
existed in the measurement of the diameter of the rod. Since,
as we have seen, the observation is satisfied by m =-02822, and
(147) gives aostlX when yl and t are constant, it is sufficient,
in order to satisfy the experiment, to increase the diameter of
the rod in the ratio of 0*2759 to 0*2822, or to suppose an error of

ON THE MOTION OF PENDULUMS.

83

only 0*0017 inch in defect to have existed in the measurement of
the diameter.

57. I proceed next to the experiments on spheres attached
to fine wires. The pendulums of this construction comprise four
1^-inch spheres, Nos. 1, 2, 3, and 4; three 2-inch spheres, Nos. 5,
6, and 7; and one 3-inch sphere, No. 66. Nos. 8 and 9 are the
same spheres as Nos. 5 and 7 respectively, swung by suspending
the wire over a cylinder instead of attaching it to a knife-edge
apparatus. As this mode of suspension was not found very satis¬
factory, and the results are marked by Baily as doubtful cases, I
shall omit the pendulums Nos. 8 and 9, more especially as with
reference to the present inquiry they are merely repetitions of
Nos 5 and 7.

In the case of a sphere attached to a fine wire of which the
effect is neglected, and swung in an unconfined mass of fluid, we
have by the formulae (52)

(148),

2a being in this case the diameter of the sphere. Before employ¬
ing this formula in the comparison of theory and experiment, it
will be requisite to consider two corrections, one for the effect of
the wire, the other for the effect of the confinement of the air by
the sides of the vacuum tube.

I have already remarked at the end of Section IV., Part I.,
that the application of the formulas of Section III. to the case of
such fine wires as those used in pendulum experiments is not
quite safe. Be that as it may, these formulae will at any rate
afford us a good estimate of the probable magnitude of the
correction.

Let l be the length, a x the radius, V 1 the volume of the wire,
V the volume of the sphere, I the moment of inertia of the
pendulum, T that of the air which we may conceive dragged
by it, ff the sum of the elements of the mass of the pendulum
multiplied by their respective vertical distances below the axis
of suspension, H' the same for the air displaced, a the density
of the air. Then the length of the isochronous simple pendulum
is Iff' 1 in vacuum, and (I + F) (ff — ff ') _1 in air, and the time
of vibration is increased by the air in the ratio of F 2 H~^ to

6—2

84 ON THE EFFECT OF THE INTERNAL FRICTION OF FLUIDS

(I+iy(H—H')-*, or, on account of the smallness of er, in the
ratio of 1 to l + \ {IT 1 + nearly. Now § H'H' 1 is the

correction for buoyancy, and therefore

.(149).

We have also, if k t be the value of the function k of Section
III., Part I.,

r=kcrV (Z-f a) 2 + H' = o-V(l + a) + ... (150),

and j0T“ 1 = (l -f a) -1 very nearly. Substituting in (149), expanding
the denominator, and neglecting F t 2 , we get

U- 1 = k +

k

l

l + CJb

Now V t is very small compared with F, and it is only by being
multiplied by the large factor \ that it becomes important. We
may then, without any material error, replace the last term in the
above equation by J V x V~ l l 2 (l 4- a)~\ and if X be the length of the
isochronous simple pendulum, we may suppose l + a = X, and
replace l 2 ( Z 4- a)’ 2 by 1 — 2aX~\ since a is small compared with X.
We thus get, putting Att for the correction due to the wire,

All —S^(!-??)(*,-i).

Substituting for — 1 from (115), and for m from (147), in
which equations, however, k v a x must be supposed to be written
for k } a , expressing V v Fin terms of the diameters of the wire
and sphere, and neglecting as before a 2 in comparison with X 2 ,
we get

where

An =

(2X — 3 x 2a) (jlt

(151),

- i “ l 0 «- 4 ; v / $- 0 ' 5772 .

It is by these formulae that I have computed the correction
for the wire in the following table. In the experiments, the time
of oscillation was so nearly one second that it is sufficient in the
formulae (148), (151), aud (152) to put r= 1, and take X for the
length of the seconds’ pendulum, or 39*14 inches.

ON THE MOTION OF PENDULUMS.

85

With respect to the correction for confined space, it seems
evident that the vacuum tube must have impeded the free motion
of the air, and consequently increased the resistance experienced
by the pendulum when it was swung in air, and that the increase'
of resistance caused by the cylindrical tube must have been some¬
what less than that which would have been produced by a spheri¬
cal envelope of the same radius surrounding the sphere. The
effect of a spherical envelope has been investigated in Section II.,
Part I.; but as we are obliged at last to have recourse to estima¬
tion, it is needless to be very precise in calculating the increase of
resistance due to such an envelope, and we may accordingly
employ the expression obtained from the ordinary theory of hydro¬
dynamics. According to this theory, the increase of the factor k ,
which is due to the envelope, is equal to f a 3 (6 s — a 3 ) -1 , or f a?b~ s
nearly, when b is large compared with a. The increase due to a
cylindrical envelope whose axis is vertical, and consequently per¬
pendicular to the direction of oscillation of the sphere, may be
estimated at about two-thirds of the increase due to a spherical
envelope of the same diameter. I have accordingly taken + a 3 b ~ 3
for the correction for confined space, and have supposed 2b = 6'5
inches.

The diameter of the wire employed in the pendulums Nos. 1,
2, 3, 5, 6, and 7, is stated to have been about the 7 tyth of an inch,
and that of the wire employed with the heavy brass sphere No. 66,
about 0023 inch. The ivory sphere No. 4 was swung with a fine
wire weighing rather more than half a grain. Taking the weight
at half a grain, and the specific gravity of silver at 10*5, we have
for this wire 2a x = 0*00251 nearly. The diameters of the three
brass spheres in the following table are taken from page 447 of
Baily’s memoir. The several parts of which, according to theory,
n is composed, are exhibited separately.

The mean error in different determinations of tt for the same
sphere was about 0*01 or 0*02, and this does not include errors
arising from small errors in specific gravities, &c. Hence, if we
except the spheres Nos. 1, 2, and 4, the discrepancies between
theory and experiment are altogether insignificant. In considering
the confirmation thence arising to the theory, it must be borne in
mind that the theory did not furnish a single disposable constant,
inasmuch as \j$ j! was already determined from the experiments

86 ON THE EFFECT OF THE INTERNAL FRICTION OF FLUIDS

Baily’s results with spheres suspended by fine wires.

No. and kind

Diameter

of

sphere

2 a

Diameter

of

wire.

2 CL i

tt By theory

Por

buoy¬

ancy

Por

inertia

on

common

theory

Additional
for inertia
on account
of internal
friction

li-INCH SPHERES

"No. 1, Platina

1*44

0-01429

1

0*5

0*289

No. 2 , Lead

1*46

0*01429

1

0*5

0*285

No. 3, Brass

1*465

0*01429

1

0*5

0*284

No. 4, Ivory

1*46

0*00251

1

0*5

0*285

2-inch spheres

No. 5, Lead

2*06

0*01429

1

0*5

0*202

No. 6 , Brass

2*065

0*01429

1

0*5

0*202

No. 7, Ivory

2*06

0*01429

1

0*5

0*202

3-inch sphere

No. 66 , Brass

3*030

0*023

1

0*5

0*137

n By theory (continued)

No.

1

2

3

4

5

6
7

66

Correction
for wire

Correction
for confin¬
ed space

Total

n

By expe¬
riment

0*035

0*011

1*835

1*881

0*035

0*011

1-831

1*871

0*035

0*011

1*830

1*834

0*016

0*011

1*812

1*872

0*012

0*032

1*746

1*738

0*012

0*032

1*746

1*751

0*012

0*032

1*746

1*755

0*005

0*101

1*743

1*748

Difference

-1-0*046, or
+ 0*040, or +~ 4 V
+ 0*004, or + T J f
+0*060, or +^

— 0*008, or —

+ 0*005, or + ^
+0*009, or +i-cjrg;

+0*005, or + 3 ^

with cylindrical rods. The result obtained with the brass sphere
No. 3 happens to agree almost exactly with theory. However, as
the results obtained with this sphere exhibited some anomalies, it
seems best to exclude it from consideration. The value of tt, then,
which belongs to a 1\ inch sphere, appears to exceed by a minute
quantity the value deduced from theory. The difference is indeed
so small that it might well be attributed to errors of observation,
were it not that all the spheres tell the same tale. Thus the
error + 0 046 in the case of the platina sphere corresponds to an

ON THE MOTION OF PENDULUMS.

87

error of less than the fortieth part of a second in the observation
of an interval of time amounting to 4J hours. If the apparent
defect, amounting to about 0*04 or 0 05, in the theoretical result
be real, it may be attributed with probability to an error in the
correction for the wire. This would be no objection to the theory,
for it will be remembered that the theory itself indicated the
probable failure of the formulae generally applicable to a long
cylinder when the cylinder comes to be of such extreme fineness
as the wires employed in pendulum experiments.

58. The preceding experiments of Baily’s are the most im¬
portant for the purposes of the present paper, inasmuch as they
were performed on pendulums of simple and very different forms;
but there still remain three sets of experiments, the fourteenth,
fifteenth, and sixteenth, in which the pendulum consisted of a
combination of a sphere and a rod, so that the results can be
compared with theory. The details of these experiments being
suppressed, I have been obliged to calculate the time of oscillation
from the ordinary formulae of dynamics, but the results will no
doubt be accurate enough for the purpose required. In all the
calculations I have supposed the rod to reach up to the axis of
suspension, and have consequently added 1*55 inch (the length of
the shank of the knife-edge apparatus) to the length of the rod,
and have added to the weight of the rod a quantity bearing to
the whole weight the ratio of 1*55 inch to the whole length.

In the case of the spheres attached to the ends of the rods
(sets 14 and 16) the process of calculation is as follows. Let l be
the length of the rod increased by 1*55 inch, W t its weight,
increased as above explained, a the radius and W the weight of
the sphere, X the length of the isochronous simple pendulum.
Then supposing the masses of the rod and sphere to be respec¬
tively distributed along the axis, and collected at the centre, which
will be quite accurate enough for the present purpose, and putting
a for the ratio of a to l y we have by the ordinary formula

. iTr i + (l+a)-TF.

2 W~+ (1 + a) F

(153),

whence r, the time of vibration, is known. The formula (148)
then gives k, which applies to the sphere, and (147) gives ut, the a

88 ON THE EFFECT OF THE INTERNAL FRICTION OF FLUIDS

in this formula being the radius of the rod, from whence which
applies to the rod, may be got by interpolation from the table in
Part I. Let A k, A/c, be the corrections which must be applied to
k, k on account of the confined space of the vacuum apparatus,
and let S u S be the specific gravities of the rod and sphere respec¬
tively; then we get by means of the formulae (149), (150)

n-1 =

W W

} (k, + A&,) - 1 + (1 + a) 2 (Jo + A k)

i + (1 + af W

jy,+(i+«)y

x —w W '

i^ + a + a)-^

S

.(154).

The first of the two factors connected by the sign x in this equa¬
tion is equal to <r~ l T 7" 1 , and if we want to calculate the weight of
air which we must conceive attached to the centre of gyration of
the pendulum in order to allow for the inertia of the air, we have
only to multiply the factor just mentioned by <r and by the weight
of the whole pendulum. The following table contains the com¬
parison of theory and experiment in the case of the 14th set. The
rods here mentioned are the same as those which composed the
pendulums Nos. 21, 43, and 44, and the spheres are the three
brass spheres of Nos. 3, 5, and 66. It appears from p. 432 of
Baily’s paper that his results are all reduced to a standard pressure
and temperature, on the supposition that the effect of the air on
the time of vibration is proportional to its density. The theory of
the present paper shews that this will only be the case if jj! be
constant, which however there is reason for supposing it to be
when the pressure alone varies. Be that as it may, no material
error can be produced by reducing the observations in this way,
because the difference of density in any pair of experiments did
not much differ from the density of air at the standard pressure
and temperature. The standard pressure and temperature taken
were 29*9218 inches of mercury and 32° F., and the assumed
specific gravity of air at this pressure and temperature was the
l-770th of that of water, so that in the calculations from theory it
is to be supposed that cr~ l = 770.

If w be the weight of the whole pendulum, w f that of the air
which we must suppose attached to the pendulum at its centre of

ON THE MOTION OF PENDULUMS.

89

gyration in order to express the effect of the inertia of the air, 8
the vibrating specific gravity of the pendulum, the effects of buoy¬
ancy and inertia are as crS~ l to ww~ x \ hut they are also as 1 to
n — 1, according to the definition of the factor tt, and therefore

w’ = (n — 1) .(155),

a formula which may be employed to calculate w' when n is
known.

Baily’s results with spheres at the end of long rods.

NTo. 45 - l|-inch sphere with copper rod.

No. 46 —2-inch sphere with ditto.

No. 47 - 3-inch sphere with ditto.

No. 48 — 1 J-inch sphere with brass rod.

No. 49 -2-inch sphere with ditto.

No. 50 -3-inch sphere with ditto.

No. 51 -1^-inch sphere with steel rod.

No. 52 —2-inch sphere with ditto.

No. 53 -3-inch sphere with ditto.

No.

i

Value of n

Weight of adhesive air, in grains

By

theory

By expe¬
riment

Difference

By

theory

By expe¬
riment

i

Difference

45

2*525

2-458

- 0-067, or - Jg-

4-863

4-564

— 0*299, or - T \ f

46

2-202

2-234

+ 0*032, or +g *<7

5-005

5-076

+ 0*071, or +ts!q

47

1-957

1-873

— 0*084, or — ;>q

7-071

6-425

— 0*646, or —

48

2-375

2-356

-0-019, or - r + f

1-447

1-417

-0*030, or -jV

49

2-060

1-982

-0*078, or -+ 3 -

2-135

1-973

-0-162, or - T b

50

1-631

1’933 ?

+ 0*302 ?

4-411

4-868?

+ 0-457 ?

51

2-099

2-344 ?

+ 0-245?

0-682

0-834 ?

+ 0-152?

52

1-920

1-793

-0-127, or

1-457

1-259

-0198, or -J

53

1-781

1-759

- 0 - 022 , or --L

3-742

3-670

— 0*072, or — Jit

With respect to the two experiments marked ? Baily remarks,
“These two experiments (with the pendulums Nos. 50 and 51)
are very unsatisfactory; and are marked as such in my journal.
It was consequently my intention to have repeated them; but the
subject was overlooked till it was too late. I should propose their
being rejected altogether.” If these two experiments be struck
out, it will be seen that the differences between theory and experi¬
ment are very small, especially when the difficulty of this set of
experiments is considered, arising from the frequency of the
coincidences with the mean solar clock.

90 ON THE EFFECT OF THE INTERNAL FRICTION OF FLUIDS

59. On account of the difficulty which Baily experienced in
obtaining accurate results with the long rods and spheres attached,
he divided the brass and steel rods near the centre of oscillation,
and after having cut off an inch from each portion inserted the
spheres where the rods had been divided. The results thus
obtained constitute the 15th set of experiments. He afterwards
removed the lower segments of the rods, and obtained the results
contained in the 16th set. I shall give the computation of the
latter set first, inasmuch as the formulae to be employed are
exactly the same as those required for the 14th set. The experi¬
ments belonging to this set in which the spheres were swung with
iron wires have already been computed under the head of spheres
attached to fine wires.

Baily's results with the spheres at the end of the short rods .

No. 60 — 1 J-inch sphere with brass rod.

No. 61 -2-inch sphere with ditto.

No. 62 -3-inch sphere with ditto.

No. 63 - 1+inch sphere with steel rod.

No. 64 - 2-inch sphere with ditto.

No. 65 —3-inch sphere with ditto.

Value of n

Weight of adhesive air

No.

theory

By expe¬
riment

Difference

By

theory

_J

By expe¬
riment

Difference

60

2-149

2*198

+0-049, or + j 5 .

1*011

1*047

+0*036, or + ^

61

1-879

1*901

+0-022, or +

1*619

1*513

— 0*106, or --

62

1-787

1*830

+0-043, or +

3*970

4*202

+ 0*232, or +^

63

1-960

1*904

-0-056, or - ;j C

0*570

0*537

-0*033, or --

64

1-796

1*785

-0-011, or -ij 3

1*239

1*227

-0*012, or --

65

1-758

1*779

+ 0-021, or + a V

3*609

3*720

+0*111, or +,

Here again the differences between theory and experiment are
extremely small. In the case of the pendulum No. 61, Baily’s
two results 1*901 and 1*513 appear to be inconsistent, as not
agreeing with the formula (155).

60. The following table contains the values of r, /c, and k x
deduced from the given data, and employed in the calculations of
which the results are contained in the two preceding tables. It is
added, partly to facilitate a comparison of the circumstances of
the different experiments, partly to assist in the re-computation

ON THE MOTION OF PENDULUMS.

91

of any of the experiments, or the detection of any numerical error
which I may have committed. I may here observe that I have
not, generally speaking, re-examined the calculations, except
where an error was apparent, but that each step requiring ad¬
dition, subtraction, multiplication, or division, was checked im¬
mediately after it was performed. I have not thought it requisite
to check in this manner the taking of logarithms or antilogarithms
out of a table.

Values of r, k, and k x employed in the calculation of the
theoretical results employed in the two preceding tables.

(For the description of the pendulums Nos. 45 to 53, see p. 89.)

Long rods

Short rods

1

No. j

T

k

No.

T

k

h

45

1*090

t

0-7968

1-951

46

1T58

0-7170

1-981

47

1-227

0*6523

2-010

48

1-155

0-8055 ;

3*222

60

0-9517

0-7772

3-012

49

1-198

0-7207 !

3*264

61

0-9806

0-7005

3-042

50

1-222

0-6520

3-288

62

0-9982

0-6373

3-062

51

1-190

0-8099

7-272

63

0-9868

0-7824

6-649

52

1-199

0-7208

7-299

64

0-9954

0-7021

6-679

53

1-231

0-6525

7-396

65

1-0030

0-6377

6*714

The corrections for confined space employed are, for the
spheres, (A/c), 0*0115, 0*0321, 0*1013; and for the rods, (A lc t ),
0*009, 0*002, 0*000. These corrections are to be added to the
values of k , k x given in the preceding table before going on with
the calculation.

61. In the 14th set of experiments, the weight of adhesive
air due to the spheres alone has been computed by Baily by
subtracting from the whole weight, as given by observation, the
weight due to the rods as given by the 13th set of experiments,
taking account of the change of weight corresponding to the
change in the position of the centre of gyration, the point at
which the air is supposed to be attached. According to theory,
this process is not legitimate, inasmuch as the weight dragged by
a rod is a function of the time of vibration, which is altered when
a sphere is attached to the end of the rod. But in the 15th set

92 ON THE EFFECT OF THE INTERNAL FRICTION OF FLUIDS

of experiments the spheres did not materially affect the time of
vibration, inasmuch as they were inserted nearly at the centre of
oscillation of the rods, and therefore in this case the process is
legitimate. Accordingly, I think it is a sufficient comparison
between theory and experiment in the case of the 15th set, to
compare the weights of air due to the spheres alone, as calculated
by Baily, with the weights calculated according to the theory of
this paper with the assumed value of V g !. I have, exhibited
separately the weight corresponding to the correction for confined
space, in order to enable the reader to form an estimate of the
extent to which the results may be affected by the uncertainty
relating to the amount of this correction.

Weights of air dragged by the spheres alone , as deduced from
Baily's results with the spheres at the centre of oscillation of the
long rods .

By Theory

ipinch

sphere

2-inch

sphere

3-inch

sphere

In free air

Additional for confined space

0-431

0-006

1-060

0-048

3-002

0-476

Total

0-437

1-108

3-478

Difference, theory and experiment, as decimal

-0-012

+ 0-001

-0101

By Experiment

lj-inch

sphere

2-inch
sphere

3-incli

sphere

From experiments with brass rod

From experiments with steel rod

0-446

0-405

1-180

1-039

3-382

3-371

Mean

0*425

1-109

3-377

Difference, as fraction of the whole

_ i

+ 1 Am

~:h

62. I pass now to Bessel’s experiments described in his
memoir entitled Untersuchungen uber die Lange des einfachen
SeJcundenpendels, which is printed among the memoirs of the
Academy of Sciences of Berlin for the 3 ear 1826. The object of

ON THE MOTION OF PENDULUMS.

93

this memoir was to determine the length of the seconds' pendulum
by a new method, which consisted in swinging the same sphere
with wires of two different lengths, the difference of lengths being
measured with extreme precision. In the calculation, the absolute
length of the simple pendulum isochronous with either the long
or the short compound pendulum was regarded as unknown, but
the difference of the two as known, and this difference, combined
with the observed times of oscillation, is sufficient for the de¬
termination of the quantity sought. Nothing more would have
been required if the pendulums had been swung in a vacuum;
but inasmuch as they were swung in air, a further correction was
necessary to reduce the observations to a vacuum. Since it is
necessary to take into account the inertia of the air, as well as its
buoyancy, in reducing the observations to a vacuum, Bessel sought
to determine by experiment the value of the factor k, of which the
meaning has been already explained. The value of this factor, as
Bessel remarked, will depend upon the form of the body; but he
does not seem, at least in his first memoir, to have contemplated
the possibility of its depending on the time of oscillation, and
consequently he supposed it to have the same value for the long
as for the short pendulum. When the factor k is introduced, the
equation obtained from the known difference of length of the two
simple pendulums contains two unknown quantities, namely k,
and the length of the seconds’ pendulum. To obtain a second
equation, Bessel made another set of experiments, in which the
brass sphere was replaced by an ivory sphere, having as nearly as
possible the same diameter. The results obtained with the ivory
sphere furnished a second equation, in which k appeared with a
much larger coefficient, on account of the lightness of ivory
compared with brass. The two equations determined the two
unknown quantities.

Let \ be the length of the seconds’ pendulum, t 1} t 2 the times
of oscillation of the brass sphere when swung with the short wire
and long wire respectively, l u l 2 the lengths of the corresponding
simple pendulums, corrected for everything except the inertia of
the air, m the mass of the sphere, the mass of the fluid dis¬
placed; then

94 ON THE EFFECT OF THE INTERNAL FRICTION OF FLUIDS

or, since m t is so small that we may neglect m x 2 ,

The long pendulum furnishes a similar equation, and the result
obtained from the brass sphere is

i-S*)“W, . (156),

since l 2 — l t is the quantity which is regarded as accurately known.
The ivory sphere in like manner furnishes the equation

MC-C)(i-^H'*- z, ‘.( 157 )>

where the accented letters refer to that sphere. The equation
for the determination of k results from the elimination of A
between the equations (156) and (157).

Now, according to the theory of this paper, the factor Jc has
really different values for the long and short pendulums. Let k x
refer to the short, and k 2 to the long pendulum with the brass
sphere, k-l to the short, and k 2 to the long pendulum with the
ivory sphere. Then

m ,

Xt.f 1

m

k)=k.

and therefore

m v V m

M,

....(158).

In the equation resulting from the elimination of A between
(156) and (157), let the values of l 2 — l x and got from (158)

and the similar equation relating to the ivory sphere be sub¬
stituted. The result is

This equation is of the form

P + + Rm* = P' + Q'm 1 + R'm^

and P = P', and Rm*, R'm* may be neglected, so that the

ON THE MOTION OF PENDULUMS.

95

equation is reduced to Q = Q'. It is now no longer necessary to
distinguish between t 2 and t 2 , and between t t and t', which may
be supposed equal. Also m : m ':: S : S', where S, S' are the
specific gravities of the brass and ivory spheres respectively.
Substituting in the equation Q = Q', and solving with respect to
k, we get

/ _ C (SK - s %) - *i 2 - S'K)

(V-WS-S')

(159).

This equation contains the algebraical definition of that
function k of which the numerical value is determined by com¬
bining, in Bessel’s manner, the results obtained with the four
pendulums. Since the equation is linear so far as regards k, k 1 ,
&c., we may consider separately the different parts of which these
quantities are composed, and add the results. For the part which
relates to the spheres, regarded as suspended by infinitely fine
wires, we have k' 2 = k 2 and k\=k 1 , since the radii of the two
spheres were equal, or at least so nearly equal that the difference
is insensible in the present enquiry. We get then from (159)

.(160),

which gives

k-k.

k /i* 2 _ k^ k^

.(161).

Since t. 2 >t 1 and k 2 >k 1} the equations (161) shew that the
value of k determined by Bessel’s method is greater than the factor
which relates to the short pendulum, which was a seconds’ pen¬
dulum nearly, and even greater than that which relates to the
long pendulum, as has been already remarked in Art. 6.

If k s be the factor relating to either sphere oscillating once in
a second, and if the effect of the confinement of the air be
neglected, we have from the formula (148)

k t ~~ i : i : : t x : t 2 : 1,

and in Bessel’s experiments 4 = 1*001, 4 = 1*721, 2a = 2*143 in
English inches. We thus get from either of the equations (160)
or (161), on substituting 0*116 for hjyt, k = 0*786. The value of
the factor k s , which relates to the sphere of the same size, swung
as a seconds’ pendulum, is only 0*694, and k x may be regarded as
equal to k s . The formula (148) gives k 2 — 0*755.

96 ON THE EFFECT OF THE INTERNAL FRICTION OF FLUIDS

63. We have next to investigate the correction for the wire.
The effect of the inertia of the air set in motion by the wire was
altogether neglected by Bessel, and indeed it would have been
quite insensible had the parts of the correction for inertia due to
the wire and to the sphere, respectively, been to each other in
nearly the same ratio as the parts of the correction for buoyancy.
Baily, however, was led to conclude'from his experiments that the
effect of the wire was probably not altogether insignificant, and
the theory of this paper leads, as we have seen, to the result that
the factor n is very large in the case of a very fine wire.

The ivory sphere in Bessel’s experiments was swung with a finer
wire than the brass sphere. It was for this reason that I did not
from the first suppose &/ = &, and k 2 ~k 2 . Let A k, A k x &c. be
the corrections due to the wire. The values of A Jc 19 A k 2 , A A/, Ak 2 \
may be got from the formula (151), in which it is to be re¬
membered that \ denotes the length of the isochronous simple
pendulum, not, as in Bessel’s notation, the length of the seconds’
pendulum. It is stated by Bessel (p. 131), that the wire used
with the brass sphere weighed 10*95 Prussian grains in the case
of the long pendulum, and 3*58 grains in the case of the short.
This gives 7*37 grains for the weight of one toise or 72 French
inches. The weight of one toise of the wire employed with the
ivory sphere was 6*28 — 2*04 or 4*24 grains (p. 141). The specific
gravity of the wire was 7*6 (p. 40), and the weight of a cubic line
(French) of water is about 0*1885 grain. From these data it
results that the radii of the wires were 0*003867 and 0 002933
inch English. The formula (147) gives m, whence L is known
from (152). The lengths of the isochronous simple pendulums
were about 39*20 inches for the short pendulum, and 116*94 for
the long. On substituting the numerical values we get from
(151), since ^ = n x — 1 and k 2 ~U 2 — 1,

A* x = 0*0107, A k 2 = 0*0286, A k' = 0*0090, A k 2 = 0*0244.

The specific gravities of the two spheres were about 8*190 and
1*794, whence we get from (159) A/c = 0*0308, or 0*031 nearly.

The value of k deduced by Bessel from his experiments was
0*9459 or 0*946 nearly, which in a subsequent paper he increased
to 0*956. In this paper he contemplates the possibility of its
being different in the cases of the long and of the short pendulum,

ON THE MOTION OF PENDULUMS.

97

and remarks with justice that no sensible error would thence
result in the length of the seconds’ pendulum, as determined by
his method, but that the factor k would belong to the system of
the two pendulums.

The following is the result of the comparison of theory and
experiment in the case of Bessel’s experiments on the oscillations
of spheres in air.

Value of k belonging to the system of a long and
a short pendulum, as determined experimen¬

tally by Bessel . 0*956

Value deduced from theory, including the correc¬
tion for the wire, but not the correction for
confined space . 0*817

difference 4- 0*139

I cannot find that Bessel has stated exactly the distance of the
centre of the sphere from the back of the frame within which it
was swung, but if we may judge by the sketch of the whole
apparatus which is given in Plate I, and by a comparison of
figs. 2 and 3, Plate II., it must have been very small, that is to
say, a small fraction of the radius of the sphere*. If so, although
the exact calculation of the correction for confined space would
form a problem of extreme difficulty, it may be shewn from
theoretical considerations that the correction would be by no
means insensible, so that it might wholly or in part account
for the difference +0*139 between the results of theory and
observation. It is, however, not improbable, for a reason which
has been already mentioned, that the theoretical correction for the
wire is not quite exact.

64 The experiments performed by Bessel on a sphere vi¬
brating in water will be more conveniently considered after the
discussion of some experiment of Coulomb’s, to which I now
proceed. These experiments are contained in a memoir entitled

* The measurement of either of Bessel’s figures, figs. 5 or G, Plate II. gives
1*58 inch for the distance of the centre of the sphere from the surface of the
broad iron bar which formed the back of the frame, the surface of the bar being
supposed truly vertical; and the measurement of fig. 2 giving 2*06 inches for the
diameter of the sphere, it appears that the distance of the surface of the sphere
from the surface of the bar was barely equal to half the radius of the s})hero.

S. ITT. 7

98 ON THE EFFECT OF THE INTERNAL FRICTION OF FLUIDS

Experiences destinies a determiner la coherence des fluides et les
lois de leur resistance dans les niouvements tres-lents, which will
be found in the 3rd Volume of the Mimoires de VInstitute p. 246.
The experiments which I shall first consider are those which
relate to the oscillations of disks suspended in water with their
planes horizontal. In these experiments the disk operated upon
was attached to the lower extremity of a vertical cylinder of
copper, not quite half an inch in diameter, the axis of which
passed through the centre of the disk. The cylinder was sus¬
pended by a fine wire attached to its upper extremity. The
under portion of the cylinder, together with the attached disk,
were immersed in water, the disk at the bottom of the cylinder
being immersed to the depth of 4 or 5 centimetres below the
surface. The upper portion carried a horizontal metallic graduated
disk, by means of which the arc of oscillation could be read off,
and which, on account of its size and weight, mainly determined
the inertia of the system, so that the time of oscillation in the
different experiments was nearly the same. The observations
were taken as follows. The whole system was turned very slowly
round by applying the hands of the graduated disk, taking care
not to derange the vertical position of the suspending wire. The
arc through which the system had been turned was read by
means of the graduation, or rather the system was turned through
an arc previously fixed on ; the system was then left to itself, and
the arc again read off to a certain number of oscillations. Thus it
was the decrement of the arc of oscillation that was observed; the
time of oscillation was indeed also observed, but only approxi¬
mately, for the sake of determining a subsidiary quantity required
in the calculation. Indeed, it will be easily seen that the experi¬
ments were not adapted to determine the effect of the fluid on
the time of oscillation. The decrement of arc so determined had
to be corrected for the effect of the imperfect elasticity of the
wire, and of the resistance of the air against the graduated disk,
and of the water against the portion of the copper cylinder
immersed. The amount of the correction was determined by
repeating the observation when the lower disk had been removed.

It appeared from the experiments, first, that with the same
disk immersed, the successive amplitudes of oscillation decreased
in geometric progression; secondly, that with different disks the

ON THE MOTION OF PENDULUMS.

99

moment of the resisting force was proportional to the fourth
power of the radius. From these laws Coulomb concluded that
each small element of any one of the disks experienced a
resistance varying as the area of the element multiplied by
its linear velocity. It should be observed that Coulomb was
only authorized by his experiments to assert this law to be
true in the case of oscillations of given period, inasmuch as
the time of oscillation was nearly the same in all the experi¬
ments.

Let a be the radius of the disk in the fluid, r the time of
oscillation, 9 the angular displacement of the disk, measured from
its mean position, I the moment of inertia of the whole system;
and let 1 : 1 — m be the ratio in which the arc of oscillation
is diminished in one oscillation. According to the formula (15)
we have

€ -nPt

for the factor which expresses the ratio of the arc of oscillation at
the end of the time t to the initial arc. At the end of one oscilla¬
tion t — r, and the value of the above factor is 1 — m, which is
given by observation. Putting for j3 its value, in which Mrf = J,
and nr = n r, we get

log. (1 - m) = - *y- aJ —(p.(!62).

Let T be the time of oscillation, and I 0 the moment of inertia,
when the under disk is removed: then I = J 0 t 2 T~ 2 . Also if M be
the mass and R the radius of the large graduated disk, we have
I 0 = ^MR'\ neglecting, as Coulomb did, the rotatory inertia of the
copper cylinder. Substituting in (162), we get

log e (1 - m)" 1 = 2 “*t t^p^t^TWRT^M- 1 .(163).

Let W be the weight of the disk in grammes. Then the mass of
the disk is equal to that of W cubic centimetres or 1000 W cubic
millimetres of water. Hence M = lOOOphF, a millimetre being the
unit of length. Substituting in (163), and solving with respect to
tjyj, we get

V// = 1000 x 2* Iog e 10.7T~- log 10 (1 - m)' 1 .. .(164),

and the same value of vV ought to result from different experi¬
ments.

7—2

100 ON THE EFFECT OF THE INTERNAL FRICTION OF FLUIDS

The weight of the disk is stated to have been 1003 grammes,
and its diameter 271 millimetres, and it made 4 oscillations in
91 seconds. Hence W — 1003, It= 1355, T = 22*75. The last
three factors in (164) vary from one experiment to another.
After making experiments with three disks of different radii
attached to the copper cylinder, Coulomb made another set with
nothing attached, for the purpose of eliminating the effect of the
imperfect elasticity of the wire. The following table contains the
data furnished by experiment, together with the value of s/y!
deduced from the several experiments. The latter is reduced
to the decimal of an English inch, by including 2*5952 (the
logarithm of the ratio of a millimetre to an inch) in the logarithm
of the constant part of the 2nd member of equation (164).

Determination of the value of fyf for water from Coulomb's
experiments on the decrement of the arc of oscillation of disks,
oscillating in their own plane by the force of torsion.

No.

Diameter
of disk

2 a

in millimetres

Time of
four

oscillations

4 r

log,o (1- m)- 1

Resulting
value of

. w;

m inches

1

195

97

0-0568

005519

2

140

92

0*021

0-05716

3

119

91

0*0135

0-05436

4

0

91

0*0058

In correcting the results of the first three experiments for the
imperfect elasticity of the wire, Coulomb calculated the values of
m given by the four experiments, and subtracted the value given
by the fourth from each of the others. But it is at the same time
easier and more exact to subtract the value of log(l — mf 1 given
by the fourth experiment from that given by each of the others.
For if

be the moments of two forces, each varying as the velocity,
divided by the moment of inertia, the factors by which the
initial arc of oscillation must be multiplied to get the arc at
the end of the time t , first, when the two forces act together,
secondly, when the second force acts alone, are

ON THE MOTION OF PENDULUMS.

101

£*“ (C+C ) t p—C £

c , c ,

respectively, and that, whether the time t be great or small.
Hence if we subtract the logarithm of the second factor from
that of the first we shall get the logarithm of the factor due
to the action of the first force alone. But if we put each factor
under the form 1 — m, and subtract the m of the second factor
from the m of the first, we shall not get the m due to the
first force alone, unless t be small enough to allow of our neglect¬
ing the squares of ct and ct , or at least the product ct . c't In
truth, when t = r, the quantities m are sufficiently small to be
treated in Coulomb’s manner without any material error, since the
corrected values of log (1 — m), obtained in the two ways, would
only differ in the 4th place of decimals.

The numbers given in the last column of the above table were
calculated from the formula (164), on substituting for log(l —m)" 1
the numbers found in the first three lines of the 4th column,
corrected by subtracting 0*0058. The mean of the three results is
0*05557, but the three experiments are not equally valuable for
the determination of vV* For the three numbers from which *J/jl
was deduced are 0*0510, 0*0152, 0*0077, and a given error in the
first of these numbers would produce a smaller error in vV than
that which would be produced by the same error in the second,
still more, than that which would be produced by the same error
in the third. If we multiply the three values of yV by 510, 152,
and 77, respectively, and divide the sum of the products by
510 + 152 + 77 or 739, we get 0*05551. We may then take 0*555
as the result of the experiments. Assuming fi = 0*0555 we have

log (1-7?*) -1 from experiment 0-0568 in No. 1, 0*021 in No. 2, 0-0135 in No. 3,

. from theory 0*0571 0*0206 0*0137

difference -0*0003 +0*0004 -0*0002

65. So far the accordance of the theoretical and observed
results is no very searching test of the truth of the theory. For,
in fact, the theory is involved in the result only so far as this,
that it shews that the resistance experienced by a given small
element of a disk oscillating in a given period varies as the linear
velocity; since the difference of periods in Coulomb’s experiments
was so small that the effects thence arising would be mixed up
with errors of observation. This law is so simple that it might

102 ON THE EFFECT OF THE INTERNAL FRICTION OF FLUIDS

very well result from theories differing in some essential particu¬
lars from the theory of this paper. But should the numerical
value of V/ determined by Coulomb’s experiments on disks be
found to give results in accordance with theory in totally different
cases, then the theory will receive a striking confirmation. Before
proceeding to the discussion of other experiments, there are one
or two minute corrections to be applied to the value of *J(i given
above, which it will be convenient to consider.

In the first place, the result obtained in Art. 8 is only approxi¬
mate, the approximation depending upon the circumstance that
the diameter of the revolving body is large compared with a
certain line determined by the values of . p and r. In the
particular case in which the revolving solid is a circular disk, it
happens that the approximate solution satisfies the general equa¬
tions exactly, except so far as relates to the abrupt termination of
the disk at its edge*. In consequence of this abrupt termination,
the fluid annuli in the immediate neighbourhood of the edge are
more retarded by the action of the surrounding fluid than they
would have been were the disk continued, and consequently the
resistance experienced by the disk in the immediate neighbour¬
hood of its edge is actually a little greater than that given by the
formula. I have not investigated the correction due to this
cause, but it would doubtless be very small.

In the second place, the formula (15) is adapted to an in¬
definite succession of oscillations, whereas Coulomb did not turn
the disk through an angle greater than the largest intended to
be observed, and suffer one or two oscillations to pass before the
observation commenced, but took for the initial arc that at which
the disk had been set by the hand. Probably the disk was held
in this position for a short time, so that the fluid came nearly to
rest. If so, the resulting value of vV, as may readily be shewn,
would be a little too small. For in the course of an indefinite
series of oscillations, the disk, in its forward motion, carries a
certain quantity of fluid with it, and this fluid, in consequence of
its inertia, tends to preserve its motion. Hence, when the disk,
having attained its maximum displacement in the positive direc¬
tion, begins to return, it finds the fluid moving in such a manner
as to oppose its return, and therefore it experiences a greater

See note A. at the end.

ON THE MOTION OF PENDULUMS.

103

resistance than if it had started from the same position with the
fluid at rest. In fact, it appears from the expression for G in
Art. 8, that the moment of the resistance vanishes, in passing
from negative to positive, not when the disk has reached the end
of its excursion in the positive direction, hut the eighth part of a
period earlier. Hence, had the observation commenced during a
series of oscillations, a larger initial arc would have been necessary,
to overcome the greater resistance, in order to produce, after a
given number of oscillations, the same final arc as that actually
observed. I have investigated the correction to be applied on
account of this cause, and find it to be about + 0*009, but I must
refer to a note for the demonstration, in order not to interrupt
the present discussion*. I shall assume then, in the following
comparisons, that for water

^f/jf = 0*0564,

the units being the sam® as before, namely, an English inch and a
second. That fi is independent of the pressure of the fluid, or at
least very nearly so, appears from an experiment of Coulomb’s, in
which it was found that the decrement of the arc of oscillation of
a disk oscillating in water was the same in an exhausted receiver
as under the full atmospheric pressure.

I will here mention another experiment of Coulomb’s which
bears directly on one part of the theory. On covering the disk
with a thin coating of tallow, the resistance was found to be the
same as before; and even when the tallow was sprinkled with
powdered sandstone, by means of a sieve, the increase of resistance
was barely sensible. This strikingly confirms the correctness of
the equations of condition assumed to hold good at the surface
of a solid.

66. I will now compare the formula (148) with the results
obtained by Bessel for the oscillations of the brass sphere in water,
which will be found at page 65 of his memoir. This sphere was
suspended so as to be immersed in the water contained in a large
vessel, and was swung with two different lengths of wire, the
same as those employed for the experiments in air. The times of
oscillation were 1*9085 second for the long pendulum, and 1*1078
for the short. The results are

* See note B. at the end.

104 ON THE EFFECT OF THE INTERNAL FRICTION OF FLUIDS

Long pendulum.

k y by experiment.0*648

k , by theory.0*631

Short pendulum,
0*602
0*600

difference -F 0*017 -1- 0*002

The depth to which the spheres were immersed is not stated, but
it was probably sufficient to render the effect of the free surface
small, if not insensible. The vessel was three feet in diameter,
and the water 10 inches deep, so that unless the spheres were
suspended near the bottom, which is not likely to have been the
case, the effect of the limitation of the fluid by the sides of the
vessel must have been but trifling. The agreement of theory and
observation, as will be seen, is very close.

67. In the same memoir which contains the experiments on
disks, Coulomb has given the results of some experiments in
which the disk immersed in the fluid was replaced by a long
narrow cylinder, placed with its axis horizontal and its middle
point in the prolongation of the axis of the vertical copper cylinder.
In these experiments, the arcs did not decrease in geometric
progression, as would have been the case if the resistance had
varied as the velocity; but it was found that the results of
observation could be satisfied by supposing the resistance to vary
partly as the first power, and partly as the square of the velocity.
In Coulomb’s notation, 1 : 1 — m denotes the ratio in which the
arc of oscillation would be altered after one oscillation, if the
part of the resistance varying as the square of the velocity were
destroyed. The several experiments performed with the same
cylinder were found to be sufficiently satisfied by the formula
deduced from the above-mentioned hypothesis respecting the re¬
sistance, when suitable numerical values were assigned to two
disposable constants m and p, of which p related to the part of the
resistance varying as the square of the velocity.

Conceive the cylinder divided into elementary slices by planes
perpendicular to its axis. Let r be the distance of any slice from
the middle point, 6 the angle between the actual and the mean
positions of the axis, dF that part of the resistance experienced by
the slice which varies as the first power of the velocity. Then
calculating the resistance as if the element in question belonged

ON THE MOTION OF PENDULUMS.

105

to an infinite cylinder moving with the same linear velocity, we
have by the formulae of Art. 31

dF = k'M'n ^ , where M = 7 rpcfdr, = r ^.

If G be the moment of the resistance, l the whole length of the
cylinder, we have, putting n = 7 rr wl ,

r 7rVcpa 2 l 5 dd
(r_ 12r di 1

whence

loge (1 - m)- 1 = —^T

.(165),

I being the moment of inertia.

Expressing I in terms of the same quantities as in the case of the
disk, we get from (147) and (165)

logic (! ~ m T = log 10 e. ^ ^ • m 2 //

•(166),

and gp is the weight of a cubic millimetre of water, or the 1000th
part of a gramme. The numerical values of pf, T } R , W have
been already given, but p! must be reduced from square inches to
square millimetres. The cylinders, of which three were tried in
succession, had all the same length, namely, 249 millimetres.
Their circumferences, calculated from their weights and expressed
in millimetres, were 21T, 11*2, and 0’87, and the time of four
oscillations was 92 s , 91 s , 91 s . The values of m calculated from
these data by means of the formula (147) are 0*4332, 0*2312, and
0*01796. For the first and second of these values, \\\ 2 h r may be
obtained by interpolation from the table given in Part I.; for the
third it will be sufficient to employ the second of the formulae
(115).

The following are the results:

Cylinder, No. 1. No. 2. No. 8.

m, by experiment . 0*0400 0*0260 0*0136

m, by theory . 0*0413 0'0291 0*0113

Difference — 0*0013 — 0*0031 + 0*0023

The differences between the results of theory and experiment
are perhaps as small as could reasonably be expected, when it is

106 ON THE EFFECT OF THE INTERNAL FRICTION OF FLUIDS

considered that, notwithstanding the delicate nature of the experi¬
ments, the numerical values of two constants, m and p , had to be
deduced from their results.

68. This memoir of Coulomb’s contains also a notice of a set
of experiments with disks and cylinders in which the water was
replaced by oil. The experiments with disks shewed that with a
given disk the arc of oscillation decreased in geometric progres¬
sion, and that with different disks the moments of the resistances
were as the fourth powers of the diameters. The absolute resist¬
ances were greater than in the case of water in the ratio of about
17*5 to 1. The details of Coulomb’s experiments on cylinders
oscillating in oil are entirely omitted. It is merely stated that on
making the same cylinders as before, or shorter cylinders when
the resistance was too great, oscillate in oil, it was found, con¬
formably with the results obtained with planes, that the coherence
of oil was to that of water as 17 to 1. The coherence is here
supposed to be measured by that part of the resistance which is
proportional to the first power of the velocity. On making a
rough calculation of the ratio of the resistances to cylinders oscil¬
lating in oil and in water, on the supposition that *Jp! for oil is to
\JfjL for water as 17*5 to 1, as would follow from the experiments
on disks if the difference of the specific gravities of the two fluids
be neglected, I found that the ratio in question ought to have
been somewhere about 100 to 1, instead of only 17 to 1. It
would seem from this that the theory of the present paper is not
applicable to oil; but fresh experiments would be required before
this point can be considered as established, on account of the
theoretical doubt respecting the application of the formulae of
Section III. Part I., to extremely fine cylinders, especially in cases
in which p! is large, so that m is very small. It would be interest¬
ing to make out whether what I have called internal friction is or
is not of the same nature as viscosity. Coulomb and Dubuat
apply the term viscosity to that property of water by virtue of
which certain effects are produced which have been shewn in this
paper to be perfectly explicable on the theory of internal friction;
whereas Poisson, in one of his memoirs, expressly asserts that the
terms in the equations of motion which result from what has been
called in this paper internal friction belong to perfect fluids, and

ON THE MOTION OF PENDULUMS.

107

have nothing to do with viscosity*. Poisson does not give the
slightest hint as to the grounds on which he rested his opinion.

69. I come now* to the experiments of Dubuat, which are
contained in an excellent work of his entitled Principes d’Hydrau-
lique , of which the second edition was published in 1786. The
first edition does not contain the experiments in question. Dubuat
justly remarked that the time of oscillation of a pendulum oscil¬
lating in a fluid is greater than it would be in vacuum, not only
on account of the buoyancy of the fluid, which diminishes the
moving force, but also on account of the mass of fluid which must
be regarded as accompanying the pendulum in its motion; and
even determined experimentally the mass of fluid which must be
regarded as carried by the oscillating body in the case of spheres
and of several other solids. Thus Dubuat anticipated by about
forty years the discovery of Bessel; but it was not until after the
appearance of Bessel's memoir that Dubuat’s labours relating to
the same subject attracted attention.

Dubuat’s method was as follows. Imagine a body suspended
by a fine thread or wire and swung in vacuum, and let a be the
length of the pendulum, reckoned from the centre of suspension
to the centre of oscillation. Now imagine the same body swung
in a fluid, in which its apparent weight is p, so that if P denote
the weight of fluid displaced, the true weight of the body will be
p + P- Since the moving force is diminished in the ratio of p + P
to p, if the inertia of the body were all that had to be overcome, it
would be necessary to diminish the length of the pendulum in the
same ratio, in order to preserve the same time of oscillation. But
since the mass in motion consists not only of the mass of the body
itself, but also of that of the fluid which it carries with it, the
pendulum must be shortened still more, in order that the time of
oscillation may be unaltered. Let l be the length of the pendulum
so shortened, and tt (which for the same reason as before I write
instead of Dubuat's n,) a factor greater than unity, such that
p + VlP is the weight of the mass in motion; then

l= p~?nP’ whencen = |(f-l) .(167).

* •Journal de VEcole Poly technique, Tom. xm. p. 95.

108 ON THE EFFECT OF THE INTERNAL FRICTION OF FLUIDS

Dubuat’s experiments on this subject consist of 44 experiments
on spheres oscillating in water (Tom. II. p. 236); 31 experiments
on other solids oscillating in water (p. 246); and 3 experiments
on spheres oscillating in air (p. 283). The following table con¬
tains a comparison of the formula (148) with Dubuat’s results for
spheres oscillating in water. The value of vV employed in the
calculation is 0*0564 inch English, or 0*05291 inch French.

Dubuatfs experiments on spheres oscillating in water .

it

r

calc.

obs.

diff.

Sphere of lead

Diameter 1*0113 inches

Weight in water 2102 grains

(!

v 3

1*633

1*687

1*766

1-825

1-502

1-502

1-522

1*620

—

131

*185

•244

•205

Sphere of glass

( 2

1-602

1-518

_

•084

Diameter 2*645 inches

4

1*644

1-569

-

•075

Weight in water 574 grains

U

1-676

1-598

—

•078

f 1

1-572

1-515

_

•057

Same sphere weighing

in

water

2

1-602

1-516

-

*086

2102 grains

3

1*624

1-523

—

•101

l 4

1-644

1-546

-

•098

f 1

1-572

1-537

_

*035

Same sphere weighing

in

water

2

1-602

1-523

-

•079

4204 grains

3

1-624

1-524

—

•100

l 4

1-644

1-538

-

T06

( i

1-551

1*449

_

T02

Same sphere weighing

in

water

1

1-572

1-372

-

•200

9216 grains

2

1*602

1-494

_

•108

1 3

1*624

1*494

-

•130

Sphere of wood
Diameter 4*076 inches
Weight in water 2102

grains

|i

1-566

1-581

1-593

1-614

1-507

1*547

1*547

1-567

-

*059

*034

*046

*057

( 1

1-547

1-375

_

•172

Same sphere weighing

in

water

3

1-566

1-581

1-456

1*525

-

•no

•056

4204 grains

j 4

1-593

1-557

_

•036

1-614

1*549

-

•065

( 1

1*547

1-57

+

•023

Same sphere weighing-

in

water

J 2

1*566

1-553

~

•013

9216 grains

1 3

1*581

1-59 ,

+

•009

\ 4

1-593

1-583 |

•010

ON THE MOTION OF PENDULUMS.

109

r

r

calc.

n

obs.

diff.

( 3

1*549

1-27

-*279

Another sphere of wood

Diameter 6f inches 1

Weight in water 2102 grains |

4

6

9

12

1*557

1*570

1*585

1*599

1-394

1-487

1-566

1-569

-*163

-*083

-*019

-*030

U8

1*621

1-565

-*056

Same sphere weighing in )
water 3204 grains j

•85

1*594

1-634

+ *040

( 3

1*549

1-651

+ *102

Same sphere weighing in water __

4

6

1*557

1*570

1-627

1-654

+ *070
+ *084

4204 grains

i

9

1*585

1-664

+ *079

'12

1*599

1-674

+ *075

70. If we strike out the experiments with the large sphere,
which cannot well be compared with theory for a reason which
will be explained further on, it will be observed that in seven out
of the eight groups of experiments left, the signs in the last
column are regularly minus. The preponderance of negative
errors could be destroyed by using a much smaller value of aJ/jl'
in the reduction. We have seen, however, that the value of aJ/jl
deduced from Coulomb’s experiments on the decrement of the arc
of oscillation of disks satisfied almost exactly Bessel’s observations
of the time of oscillation of a sphere about two inches in diameter
oscillating in water. The very small errors which remained in
this case had both the sign +, whereas in Dubuat’s experiments
on the 1-inch and 2-J- inch spheres, the errors, which are far
larger, have all the sign-. Since the experiments of Dubuat
and Bessel, though made under similar circumstances, do not lead
to the same result, it is of course impossible for any theory to
satisfy them both. The numbers in the last column of the pre¬
ceding table are, however, far too regular to be attributable to mere
fortuitous errors of observation. If we suppose Bessel’s results to
have been nearly exact, there must have been something in the
mode either of making or of reducing Dubuat’s experiments which
caused a tendency to error in one direction.

With respect to the reduction of the experiments it may be
observed that the length l was measured from the centre of

110 ON THE EFFECT OF THE INTERNAL FRICTION OF FLUIDS

oscillation, whereas in the formula (148) it is supposed that the
mass of which the weight is hP or (n — 1)P is collected at the
centre of the sphere. If h be the distance of the centre of the
sphere from the axis of suspension, the observed value of tt — 1
ought in strictness to be increased in the ratio of h 2 to or the
calculated value diminished in the ratio of l 2 to K 2 , before com¬
paring the results of theory and experiment. In the case of the
loaded spheres especially, the theoretical value of n would thus
be a little diminished; but except in a very few cases, in which
either l or a — l is small, the diminution is hardly worth con¬
sidering. After having been for a good while at a loss to account
for the regular occurrence of rather large negative errors, the
following occurred to me as the probable solution of the difficulty.

When a pendulum oscillates in water, the arc of oscillation
rapidly decreases; this rapid diminution forms in fact the grand
difficulty in experiments of this kind. In Dubuat’s experiments,
it will be remembered, the suspending thread was lengthened or
shortened till the time of oscillation was an exact number of
seconds, or occasionally half a second. Now, it is probable that
the observer occasionally gave the suspending thread a slight
push as the pendulum was commencing its return, in order to
keep the oscillations going for a sufficient time to allow of
tolerable precision in rendering the time of oscillation equal to
what it ought to be. If so, these pushes would slightly accelerate
the oscillations, and therefore cause the length of thread fixed
on by observation to be a little too great, which would make the
effect of the water in retarding the oscillations appear a little too
small. On inspecting the table of differences, it may be observed
that sometimes when the same sphere differently loaded is
swung in the same time as before, the numbers in the table of
differences are altered more than appears to be attributable to
merely fortuitous errors of observation. This accords very well
with the conjecture just mentioned, and seems difficult to account
for in any other way, inasmuch as everything relating to the fluid
must have been almost exactly the same in the two cases.

The occurrences of positive differences in the case of the large
wooden sphere may be accounted for by the limitation of the fluid
mass by the sides and bottom of the vessel, and by the free
surface, which, except in the case of very short oscillations, would

ON THE MOTION OF PENDULUMS.

Ill

have much the same effect as a rigid plane, inasmuch as it would
be preserved almost exactly horizontal by the action of gravity.
The vessel which contained the water was 51 inches long and
17 broad, the water was 14 inches deep, and the spheres were
plunged to about 8 inches below the surface, so that the effect
of the confinement of the fluid mass would have been quite
sensible in the case of such large spheres. If it be objected that
the same sphere gave negative differences in the case of the first
group of experiments, it must be observed, that when the apparent
weight of so large a sphere was only 2102 French grains, the
resistance would quickly have caused the oscillations to subside if
an extraneous force had not frequently been applied.

71. In Dubuat’s experiments on spheres oscillating in air, the
lightness of the fluid was compensated by the extreme lightness of
the spheres, which were composed, the first two of paper, and the
third of goldbeater's skin. In the following table the diameter
2a of the sphere is expressed in French inches. The value of \]g!
employed in the reduction is the same as was before used in the
reduction of observations made in air, namely 0*116 inch English,
or 0*1088 inch French.

Lubuat’s experiments on light spheres oscillating in air.

No.

2 a,

T

ii

calc.

it

obs.

Dill’.

337

4-0416

1-51

1-61

1-51

— o-10

338

6-625

1*84

1-57

1-63

+0-06

339

17-25

3*625

1-53

1-54

+ 0-01

The differences certainly appear very small when the delicacy
of the experiments and the simplicity of the apparatus employed
are considered.

72. The only comparison yet made in this section between
theory and observation in the case of pendulum experiments,
consists in comparing the observed times of vibration with the
results calculated with an assumed value of *Jg!. But according
to theory we ought to be able, without assigning a particular value
to any new disposable constant, to calculate the rate of decrease
of the arc of vibration, I have not met with any experiments

112 ON THE EFFECT OF THE INTERNAL FRICTION OF FLUIDS

made with a view of investigating the decrease in the arc of
vibration in the case of extremely small vibrations, such as those
employed in pendulum experiments. The experiments of Newton
and others, in which the arc of vibration was so large that the
resistance depended mainly on the square of the velocity, would
be quite useless for my purpose. The pendulum experiments of
Bessel and Baily contain however the requisite information, or
at least some portion of it, for the arcs are registered for the sake
of giving the data for calculating the small reduction to in¬
definitely small vibrations.

In Bessel’s experiments the arc is registered for the end of
equal intervals of time during the motion. The number of such
registrations in one experiment amounts in some cases to eleven,
and is never less than three. So far the observations are just
what are wanted; but there are other causes which prevent an
exact comparison between theory and experiment. In the first
place the spheres were swung so close to the back of the frame
that the increase of resistance due to the confinement of the air
must have been very sensible. In the second place the effect
of the wire must have been very sensible, especially in the case of
the long pendulum. For the table of Section III. Part I., shews
that for the wire (for which m is very small) the value of lc is
much larger than that of lc, whereas for spheres of the size of
those employed, when the time of oscillation is only one or two
seconds, lc' is a good deal smaller than lc. Hence, if the formulae
of that section applied to such fine wires, the effect of the wire
on the arc of vibration would be much greater than its effect on
the time of vibration, and therefore would be quite sensible. But
it has been shewn in Section IV., that the effect of the wire in
diminishing the arc of vibration is probably greater than would be
given by the formula, and therefore the uncertainty depending on
the wire is likely to amount to a very sensible fraction of the
whole amount. Again, since Bessel’s experiments were all made
in air, no data are afforded whereby to eliminate the portion of
the observed result which was due to friction at the point of
support, imperfect elasticity of the wire, or gradual dissipation of
ms vim by communication of motion to the supporting frame.
Moreover in the case of the long pendulum the observations were
made with rather too large arcs, for the law of the decrease of the

ON THE MOTION OF PENDULUMS.

113

arc of vibration deviated sensibly from that of a geometric pro¬
gression. In Baily’s experiments, only the initial and final arcs
are registered, and not even those in the case of the additional
experiments” Hence these experiments do not enable ns to
make out whether it would be sufficiently exact to suppose the
decrease to take place in geometric progression. Moreover, the
final arc was generally so small, that a small error committed in
the measurement of it would cause a very sensible error in the
rate of decrease concluded from the experiment. For these
reasons it would be unreasonable to expect a near accordance
between the formulae and the results of the experiments of Bessel
and Baily. Still, the formulae might be expected to give a result
in defect, and yet not so much in defect as not to form a large
portion of the result given by observation. On this account it
will not be altogether useless to compare theory and observation
with reference to the decrement of the arc of vibration.

73. Let us first consider the case of a sphere suspended by a
fine wire. Let the notation be the same as was used in in¬
vestigating the expression for the effect of the air on the time of
vibration, except that the factors k\ 1c' come in place of k, k x .
Considering only that part of the resistance which affects the arc
of vibration, we have for the portions due respectively to the
sphere and to the element of the wire whose length is ds , and
distance from the axis of suspension s ,

k'M'n(l + a)j t> k'^'-ds.ns

and if we take the moment of the resistance, and divide by twice

d6

the moment of inertia, the coefficient of in the result, taken

negatively, and multiplied by t , will be the index of e in the
expression for the arc. Hence if a 0 be the initial arc of vibration,
and a t the arc at the end of the time t

l°oe a 0 — loge a t =

k’M' (l + a f +1 ki'M 'l*
A1(1 + af + \Mj

7 rt
2 T

...(108),

M (l + of being as before taken for the moment of inertia of the
sphere, which will be abundantly accurate enough. If then we
put l for the Napierian logarithm of the ratio of the are at the

L

s. hi.

114 ON THE EFFECT OF THE INTERNAL FRICTION OF FLUIDS

beginning to the arc at the end of an oscillation, we must put t = r
in (168), whence, neglecting the effect of the wire, we obtain

Y 7 rk' cr ^

= T ~'S .

If now A y be the correction to be applied to k f in this
formula on account of the wire, since k\ k / are combined together
in the expression for the arc just as k, k x in the expression for the
time, we get

Afc' = ^AA .(170),

and the approximate formulae (115) give

Afc' = — — A&.(171),

7T

whence the numerical value of A Id is easily deduced from that of
Ak, which has been already calculated. We get also from (52)

# = (172),

whence k' may be readily deduced from k, which has been already
calculated.

74. Before comparing these formulae with Bessel's ex¬
periments, it will be proper to enquire how far the latter are
satisfied by supposing the arcs of oscillation to decrease in
geometric progression. In Bessel’s tables the arc is registered in
the column headed fi. This letter denotes the number of French
lines read off on a scale placed behind the wire, and a little above
the sphere, and is reckoned from the position of instantaneous
rest of the wire on one side of the vertical to the corresponding
position on the other side. The distance of the scale from the
axis of suspension being given, as well as the correction to be
applied to /i on account of parallax, the arc of oscillation may be
readily deduced. However, for our present purpose, any quantity
to which the arc is proportional will do as well as the arc itself,
and fjb, though strictly proportional to the tangent of the arc, may
be regarded as proportional to the arc itself, inasmuch as the
initial arc usually amounted to only about 50' on each side of the
vertical.

Now we may form a very good judgment as to the degree of
accuracy of the geometric formula by comparing the arc observed

ON THE MOTION OF PENDULUMS.

115

in the middle of an experiment with the geometric mean of the
initial and final arcs. I have treated in this way Bessels ex¬
periments, Nos. 1, 2, 3, 4, and 5. Each of these is in fact a group
of six experiments, four with the long pendulum and two with the
short, so that the whole consists of 20 experiments with the
long pendulum, and 10 with the short. In the case of the long
pendulum, the observed value of p regularly fell short of the
calculated value, and that by a tolerably constant quantity. The
mean difference amounted to 0*688 line, and the mean error in
this quantity to 0T09. This mean error was not due entirely to
errors of observation, or variations in the state of the air, &c., but
partly also to slight variations in the initial arc, larger differences
usually accompanying larger initial arcs. The initial arc usually
corresponded to = 39 or 40 lines, and the final to /jl = 15 or 16
lines. In the case of the short pendulum, the differences in 8
cases out of 10 had the same sign as before. The mean difference
was 0*025, and the mean error 0*043. The arcs of oscillation
were nearly the same as before; but inasmuch as the axis of
suspension was nearer to the scale than before, the initial value of
fju was only about 12 or 13 lines, and the final value about 7
lines. When the results of some of the experiments were laid
down on paper, by abscissae taken proportional to the times and
ordinates to the logarithms of fju, it was found that in the case
of the long pendulum the line so drawn was decidedly curved, the
concavity being turned toward the side of the positive ordinates.
The curvature of the line belonging to the short pendulum could
hardly be made out, or at least separated from the effects of errors
of observation. The experiments 9, 10, 11, having been treated
numerically in the same way as the experiments 1—5, led to
much the same result. In the 16 experiments with the ivory
sphere and short pendulum contained in the experiments Nos. 12,
13, 14, and 15, the excess of the calculated over the observed
value of fi was more apparent, the mean excess amounting to
0 129. The reason of this probably was, that the observations
with the ivory sphere were made through a somewhat wider
range of arc than those with the brass sphere.

It appears then that at least in the case of the long pendulum
a correction is necessary, in order to clear the observed decrease
in the arc of oscillation from the effect of that part of the

8—2

116 ON THE EFFECT OF THE INTERNAL FRICTION OF FLUIDS

resistance which increases with the arc more rapidly than if it
varied as the first power of the velocity, and so to reduce the
observed rate of decrease to what would have been observed in
the case of indefinitely small oscillations.

75. In Coulomb’s experiments it appeared that the resistance
was composed of two terms, one involving the first power, and the
other the square of the velocity. If we suppose the same law to
hold good in the present case, and denote the amplitude of oscilla¬
tion at the end of the time t , measured as an angle, by a, we shall
obtain

~ = -Aa-Ba* . (173),

where A and B are certain constants. We must now endeavour
to obtain A from the results of observation. Since the substitu¬
tion for a of a quantity proportional to a will only change the
constant B in (173), and the numerical value of this constant is
not required for comparison with theory, we may substitute for a
the number of lines read off on the scale as entered in Bessel’s
tables in the columns headed /jl.

I have employed four different methods to obtain A from the
observed results. The one I am about to give is the shortest of
the four, and is sufficiently accurate for the purpose.

The equation (173) gives after dividing by a

d log a . -r, sttoAs

.,< ,/i . < i74) -

Now, as has been already observed, the arcs of vibration decrease
nearly in geometric progression. If this law were strictly true, we
should have

t

where a 0 denotes the initial and a 2 the final arc, and T denotes the
whole time of observation. We may, without committing any
material error, substitute this value of a in the last term of (174).
The magnitude of the error we thus commit is not to be judged
of merely by the smallness of B. The approximate expression
(17 5) is rather to be regarded as a well-chosen formula of interpo-

ON THE MOTION OF PENDULUMS.

117

lation, and in fact T 1 log e (a 0 a 2 *) differs very sensibly from A.
Making now this substitution in (174), integrating, and after
integration restoring a in the last term by means of (175),

we get

log a = — At —

BTa

log a 2 - log a 0

+ C

(176),

G being an arbitrary constant. To determine the three constants
A, B, C, let cq be the arc observed at the middle of the experiment,
apply the last equation to the arcs a 0 , a,, a 2 , and take the first and
second differences of each member of the equation. Let A, denote
the sum of the two first differences, so that Ais the same thing
as T. Then we may take for the two equations to determine A
and B

\ log^-AAf-

BAJ. Vq ,
A t log « 0 ’

A 2 log a 0 = —

BA ,t. A 2 « „
A 2 log a 0

Eliminating jB, and passing from Napierian to common logarithms,
which will be denoted by Log., we get

- A, Log a„ L _ A 2 Log « n . A, oq
Log e. A t t \ Aj Log a 0 . A 2 a a

(177).

If we suppose the part of

~ which does not vary as the first

Cut

power of a to be a 2 <£' (a) instead of Bol\ we shall get
way

- A, Log« 0 f _ A 2 Log a 0 . A 1 ^>(a 0 )|
Log e. A x t \ A, Log a 0 . A 2 ^> (a 0 )J

in the same

.( 178 ).

76. I have not attempted to deduce evidence for or against
the truth of equation (173) from Bessel's experiments. The ap¬
proximate formula (175) so nearly satisfied the observations, that
almost any reasonable formula of interpolation which introduced
one new disposable constant would represent the experiments
within the limits of errors of observation. It may be observed,
that the factor outside the brackets in equations (177) and (178)
is the first approximate value of A got by using only the initial
and final arcs, and supposing the arcs to decrease in geometric
progression. In the case of the long pendulum, the value of A,
corrected in accordance with the formula (178), would be very
sensibly different according as we supposed <j> (a) to be equal to
Ba, in which case (178) would reduce itself to (177), or equal to

118 0N THE EFFECT OF THE INTERNAL FRICTION OF FLUIDS

Bo?. In the case of the long pendulum with the brass sphere, the
corrected value of A, deduced from the formula (177), was equal
to about 0*77 of the first approximate value.

I have not considered it necessary to go through all Bessels
experiments, as it was not to be expected that the formula should
account for the whole observed decrement. I have only taken
four experiments for each kind of pendulum, namely, I. a, b, e, and
/ for the long pendulum with the brass sphere; I. c and d and II.
c and d for the short pendulum with the brass sphere; XII. a, 6,
c, and d for the long pendulum with the ivory sphere, and XII. a\
b\ d ', and d' for the short pendulum with the ivory sphere. The
formula (177) gave the following results.

First case,

Log e. t A = *0000759 ; mean error = ’0000020.

Second case,

Log e. rA = *0000504; mean error = *0000075.

Third case,

Log e. A = *000631; mean error = *000046.

Fourth case,

Log e. A = *000167 ; mean error = *000074.

Now l — tA, and therefore, to get the values of I deduced from
experiment, it will be sufficient to divide the numbers above
given by the modulus of the common system of logarithms. The
theoretical value of I will be got from (169), if we add to k f the
correction A k' depending upon the wire. The following are the
results:

long p.

short p.

long p.

short p.

1000000 l for sphere alone in
an unlimited mass of fluid,

brass s.

brass s.

ivory s.

ivory s.

by theory ..

67

50

298

222

additional for wire .

27

9

114

39

94

59

412

261

1000000 l by experiment...

175

116

1453

384

It appears then that the calculated rate of decrease of the arc
amounts on the average to about half the rate deduced from
observation. This is about what we might have expected, con¬
sidering the various circumstances, all tending materially to

ON THE MOTION OF PENDULUMS.

119

augment the rate of decrease, which were not taken into account
©

in the calculation.

77. Of Baily’s pendulums I have compared the following with
theory in regard to the decrement of the arc of vibration. No. 1
(the 1^-inch platina sphere), experiments 1 to 8 ; No. 3 (the brass
lj-inch sphere), experiments 9 to 16; No. 6 (the 2-inch brass
sphere), experiments 33 to 40; No. 21 (the 0'410 inch long copper
cylindrical rod), experiments 109 to 112; and Nos. 35—38 (the
1-^-inch long brass tube), experiments 167 to 174. I have not
thought it worth while to compute the results obtained with
the other lj-inch and 2-inch spheres, inasmuch as they were
of the same size as the brass spheres, and moreover the obser¬
vation of the decrement of the arc was not the object Baily had
in view in making the experiments. The 3-inch sphere, and all
the other cylindrical rods and combinations of cylindrical rods and
spheres, belong to the “ additional experiments ” for which the
arcs are not given.

The mode of performing the calculation will best be explained
by an example. Take, for instance, the pair of experiments Nos.
1 and 2. In No. 1 the total interval was 4*22 hours, the initial
arc was 0°*77, the iinal arc 0°*29, the mean height of the barometer
30*24 inches, and the temperature about 38£° F. The difference
of the common logarithms of the initial and final arcs is 0*424, and
this divided by the total interval gives 0*1005 for the difference of
logarithms for one hour. The second experiment, treated in a
similar way, gives 0*0352, which expresses the effect of friction at
the point of support, communication of motion to the support
itself, &c., together with the resistance of highly rarefied air at
a pressure of only 0*97 inch of mercury. Since we have reason to
believe that /j! is independent of the density, we may get the
effect of air at a pressure of 30*24-0*97 or 29*27 inches of
mercury by subtracting 0*0352 from 0*1005, which gives 0 0653.
Reducing to 29 inches of mercury for convenience of comparison,
we get 0*0649. Each pair of experiments is to be treated in the
same way. Since the temperature was nearly the same in the
experiments made with the same pendulum, we may suppose
it constant, and equal to the mean of the temperatures in the
experiments made under the full atmospheric pressure. The
experiments reduced consist of four pair for each pendulum,

120 ON THE EFFECT OF THE INTERNAL FRICTION OF FLUIDS

except No. 21, for which only two pair were performed. The
following are the results. For the 1^-inch platina sphere 0*0644,
mean error 0*0044. For the lj-inch brass sphere 0*180, mean
error 0*024. For the 2-inch brass sphere 0*094, mean error 0*013.
For the copper rod 0*486, mean error 0*113. For the brass tube
the results were 0*145, 0*363, 0*338, 0*305. Rejecting the first
result as anomalous, and taking the mean of the others, we get
0*335, mean error 0*030. To obtain I from the mean results above
given we have only to divide by 3600 times the modulus, and
multiply by t, and for the experiments with spheres we may
suppose t = 1.

The mode of calculating I from theory in the case of a sphere
suspended by a fine wire has already been explained. For the
sake of exhibiting separately the effect of the wire, I will give one
intermediate step in the calculation.

h\ for sphere alone .

1*44 inch
sphere.

0*326

1*46 inch
sphere.

0-320

2-06 inch
sphere.

0-220

A 1c', the correction for the wire

0130

0130

0045

Total, to be substituted in (169)

0456

0-450

0-265

The formula (168), which applies to a sphere suspended by a
wire, will be applicable to a long cylindrical rod if we suppose
M = 0. Hence the same formula (169) that has been used for a
sphere may be applied to a cylindrical rod if we suppose !c to
refer to the rod. For the copper rod Jc = 1*107, and for the tube
= 0*2561. The following are the results for the three spheres
and two cylinders.

1000000 1, from experiment

No. 1.

41

No. 3.

115

No. 6.
60

No. 21.

315

Nos.

35—38.

206

39

106

60

237

156

Difference.

+ 2

+ 9

0

+ 78

+ 50

It appears that the experiments with spheres are satisfied
almost exactly. The differences between the results of theory
and observation are much larger in the case of the long cylinders.
Large as these differences appear, they are hardly beyond the
limits of errors of observation, though they would probably be far
beyond the limits of errors of observation in a set of experiments

ON THE MOTION OF PENDULUMS.

121

performed on purpose to investigate the decrement of the arc of
vibration. It was to be expected beforehand that the results
of calculation would fall short of those of observation, inasmuch as
only two arcs were registered in each experiment, so that no data
were afforded for eliminating the effect of that part of the resist¬
ance which did not vary as the first power of the velocity.

78. I have now finished the comparison between theory and
experiment, but before concluding this Section I will make a few
general remarks.

When a new theory is started, it is proper to enquire how far
the theory does violence to the notions previously entertained on
the subject. The present theory can hardly be called new, because
the partial differential equations of motion were given nearly
thirty years ago by Navier, and have since been obtained, on
different principles, by other mathematicians ; but the application
of the theory to actual experiment, except in some doubtful cases
relating to the discharge of liquids through capillary tubes, and
the determination of the numerical value of the constant p!, are, 1
believe, altogether new. Let us then, in the first instance, examine
the magnitude of the tangential pressure which we are obliged by
theory to suppose capable of existing in air or water.

For the sake of clear ideas, conceive a mass of air or water to
be moving in horizontal layers, in such a manner that each layer
moves uniformly in a given horizontal direction, while, the velocity
increases, in going upwards, at the rate of one inch per second for
each inch of ascent. Then the sliding in the direction of a hori¬
zontal plane is equal to unity, and therefore the tangential pressure
referred to a unit of surface is equal to a or p!p. The absolute
magnitude of this unit sliding evidently depends only on the
arbitrary unit of time, which is here supposed to he a second.
In the case supposed, it will be easily seen that the particles
situated at one instant in a vertical line are situated at the expi¬
ration of one second in a straight line inclined at an angle of 4f>
to the horizon. Equating the tangential pressure pp lo (he
normal pressure due to a height h of the fluid, we get k ^ ,j y, ,,
being the force of gravity. Putting now <j = :{«(}, p! (()•! |i;p for
air, p = (0-0564) 2 for water, we gut h = 0T>00(m,s<> inch for air,
and h = 0-000008241 inch for water, or about the one thirty-thou-

122 ON THE EFFECT OF THE INTERNAL FRICTION OF FLUIDS

sandth part of an inch for air, and less than the one hundred-
thousandth part of an inch for water. If we enquire what must
be the side of a square in order that the total tangential pressure
on a horizontal surface equal to that square may amount to one
grain, supposing the density of air to be to that of water as 1 to
836, and the weight of a cubic inch of water to be 252*6 grains,
we get 25 feet 8 inches for air, and 1 foot 10 inches for water. It
is plain that the effect of such small forces may well be insignifi¬
cant in most cases.

79. In a former paper I investigated the effect of internal
friction on the propagation of sound, taking the simple case of an
indefinite succession of plane waves *. It appeared that the effect
consisted partly in a gradual subsidence of the motion, and partly
in a diminution of the velocity of propagation, both effects being
greater for short waves than for long. The second effect, as I
there remarked, would be contrary to the result of an experiment
of M. Biot’s, unless we supposed the term expressing this effect to
be so small that it might be disregarded. I am now prepared to
calculate the numerical value of the term in question, and so
decide whether the theory is or is not at variance with the result
of M. Biot’s experiment.

According to the expression given in the paper just mentioned,
we have for the proportionate diminution in the velocity of propa¬
gation

8ttV 2
9 A 2 V' z ’

A being the length of a wave, and V the velocity of sound. To
take a case as disadvantageous as possible, suppose A only equal
to one inch, which would correspond to a note too shrill to be
audible to human ears. Taking the velocity of sound in air at
1000 feet per second, there results for the common logarithm of
the expression above written 11*0428, so that a wave would have
to travel near 100000000000 inches, or about 1578000 miles, before
the retardation due to friction amounted to one foot. It is plain
that the introduction of internal friction leaves the theory of sound
just as it was, so far as the velocity of propagation is concerned, at
least if the sound be propagated in free air.

* Camb. Phil Trans. Vol. VIII. p. 302. [Ante, Vol. I. p. 101.]

ON THE MOTION OF PENDULUMS.

123

The effect of friction on the intensity of sound depends on the
first power of In the case of an indefinite succession of plane
waves, it appears that during the time t the amplitude of vibration
is diminished in the ratio of 1 to e ~ ct , and therefore the intensity
in the ratio of 1 to e~ 2ct , where

8t tY
C "“ '3\ 2 ’

Putting X = 1 and t = 1 we get 1 to 0*4923, or 2 to 1 nearly, for
the ratio in which the intensity is altered during one second in
the case of a series of waves an inch long. The rate of diminution
decreases very rapidly as the length of wave increases, so that in
the case of a series of waves one foot long the intensity is altered
in one second in the ratio of 1 to 0*995095, or 201 to 200 nearly.
It appears then that in all ordinary cases the diminution of inten¬
sity due to friction may be neglected in comparison with the
diminution due to divergence. If we had any accurate mode of
measuring the intensity of sound it might perhaps be just possible,
in the case of shrill sounds, to detect the effect of internal friction
in causing a more rapid diminution of intensity than would corre¬
spond to the increase of distance from the centre of divergence.

Section II.

Suggestions with reference to future experiments.

80. I am well aware that the mere proposal of experiments
does not generally form a subject fit to be brought before the
notice of a scientific society. Nevertheless, as it frequently
happens in the division of labour that one person attends more to
the theoretical, another to the experimental investigation of some
blanch of science, it is not always useless for the theorist to point
out the nature of the information which it would be most impor¬
tant to obtain from experiment. I hope, therefore, that I may be
permitted to offer a few hints with reference to experiments in
which the theory of the internal friction of fluids is concerned. I
shall omit all details, since they would properly come in connexion
with the experiments.

124 ON THE EFFECT OF THE INTERNAL FRICTION OF FLUIDS

Experiments with which the theory of internal friction in
fluids has more or less to do may be performed for either of the
following objects: first, to test still further the truth of the theory ;
secondly, to determine the index of friction of various gases,
liquids, or solutions; to investigate the dependance of the index
of friction of a gas on its state of pressure, temperature, and mois¬
ture ; or to endeavour to make out the law according to which the
index of friction of a mixture of gases depends upon the indices of
friction of the separate gases; thirdly, to measure the length of
the seconds’ pendulum, or its variation from one part of the
earth’s surface to another.

81. First object. The theory has been already put to a
pretty severe test by means of the experiments of Baily and
others. Nevertheless there are some uncertainties in the com¬
parison of theory and experiment arising from the influence of
modifying causes of which the effect could only be estimated from
theory, and yet was not so small as to be merged in errors of
observation. Moreover, experiments on the decrement of the arc
of vibration are almost wholly wanting. The following system of
pendulums, meant to be swung in air and in vacuum, would afford
a very good test of the theory.

No. 1. A 2-inch or IJ-inch sphere swung with a fine wire.'

No. 2. A very small sphere swung with the same kind of
wire.

No. 3. A long cylindrical rod, a few tenths of an inch in
diameter.

No. 4. A cylinder only three or four inches long, of the same
diameter as No- 3, swung with the same kind of wire as No. 1.

The vacuum tube ought to be of sufficient size to render the
estimated correction for confined space less than, or at most com¬
parable with, errors of observation. The vacuum apparatus used
by Col. Sabine would do very well. If the vacuum tube be not
of sufficient size, it ought to admit of removal, and to be removed
when the pendulums are swung in air.

In all the experiments the arc of oscillation ought to be care¬
fully observed several times during the motion, the observation of
the arc being quite as important for the purposes of theory as the
observation of the time. Indeed, if it should be inconvenient to

ON THE MOTION OF PENDULUMS.

125

observe the time, the observation merely of the arc would be very
valuable as a test of theory. In that case an approximate value
of the time of oscillation in air would be required.

In the system proposed, Nos. 1 and 3 are the principal pendu¬
lums, Nos. 2 and 4 are introduced for the sake of making certain
small corrections to the results of Nos. 1 and 3. No. 2 is meant
for the elimination from No. 1 of the effect of the wire, and
No. 4 for the elimination from No. 3 of the effect of the resistance
experienced by a small portion of the rod near its end. The times
of vibration of the four pendulums ought to be nearly the same,
although for that purpose slightly different lengths of wire would
be required in Nos. 1, 2, and 4.

It follows from theory that for a given pendulum the factor xi
is a function of the time of vibration. This is a result which
seems to have been hardly so much as suspected by those who
were engaged in pendulum experiments, or at most to have been
mentioned as a mere possibility*, and therefore it might be
thought advisable to verify it by direct experiment. For my own
part I regard it as so intimately connected with the fundamental
principles of the theory, that if the theory be confirmed in other
respects I think this result may be accepted on the strength of
theory alone. The direct comparison with experiment would be
inconvenient, because it would require a clock which kept excellent
time, and yet admitted of being adjusted so as to make widely
different numbers of vibrations in a day. The result could, how¬
ever, be confirmed indirectly by observing the arc of vibration, an
observation which is as easy with one time of vibration as with
another.

82. Second object According to theory, the index of friction
may be deduced from experiments either on the arc or on the
time of vibration. It must be left to observation to decide which
give the more consistent results. Should the results obtained from
the arc appear as trustworthy as those obtained from the time, it
would apparently be much the easiest way of determining // for
an elastic fluid to observe the arc, because no particular accuracy
would then be required in the observation of time. As to the

* It should be observed however that in a subsequent memoir (Astronomische
Nachrichtev, No. 223, p. 106) Bessel deduced from other experiments that the
value of k was larger for the long than for the short pendulum.

126 ON THE EFFECT OF THE INTERNAL FRICTION OF FLUIDS

form of the pendulum, a cylindrical rod would apparently be the
best if only a single pendulum were employed. The observation
of the arc seems the only practicable way of determining the in¬
fluence of temperature on the index of friction, unless the pendu¬
lum be extremely light, or unless the observer be content with the
limited range of temperature which may be procured by making
observations at different times of year. For in an apparatus
artificially heated or cooled, it would be difficult to prevent small
unknown variations of temperature, which would cause variations
in the rate of vibration, in consequence of the expansion and con¬
traction of the pendulum; and these variations would vitiate the
result of the experiment, so far as the time of vibration is con¬
cerned, because the effect of the gas on the time of vibration is
deduced from the small difference between two large quantities
which are directly observed. But the effect of the gas on the arc
of vibration produces by far the greater part of the whole diminu¬
tion observed, and therefore small fluctuations of temperature
would not be of much consequence, except so far as they might
occasion gentle currents; and even then would not be very im¬
portant, because the forces thence arising would not be periodic,
and dependent upon the phase of vibration of the pendulum.

The grand difficulty which besets the observation of the time
of vibration of a pendulum oscillating in a liquid consists in the
rapidity with which the oscillations subside. The best form of a
pendulum to oscillate in a liquid would be a sphere suspended by
a fine wire. The vessel containing the liquid and the sphere
immersed in it ought to be so large as to render the correction for
confined space insensible. But the index of friction of a liquid
would probably be better determined by experiments more of the
nature of those of Coulomb, or perhaps by the slow discharge of
liquids through narrow tubes.

Among the gases for which // ought to be determined experi¬
mentally should be mentioned coal-gas, on account of the practical
application which it appears possible to make of the result to the
laying down of gas-pipes. The calculation of the resistance in a
circular pipe is very simple, and is given in Art. 9 of my former
paper. According to the equations of condition assumed in the
present paper we must pub U-0 } U denoting in that article the
velocity close to the surface. It appears that the pressure spent

ON THE MOTION OF PENDULUMS.

127

in overcoming friction varies as the mean velocity divided by the
square of the diameter of the pipe, or as the rate of supply divided
by the fourth power of the diameter. This goes on the supposition
that the motion is sufficiently slow to allow of our neglecting the
pressure which may be spent in producing eddies, in comparison
with that spent in overcoming what really constitutes internal
friction.

83. Third object With respect to experiments for determin¬
ing the length of the seconds’ pendulum, the theory of internal
friction rather enables us to calculate for certain forms of pendulum
the correction due to the inertia of the air than points out any
particular mode of performing the experiments. Even the ordinary
theory of hydrodynamics points out the importance of removing
all obstacles to the free motion of the air in the neighbourhood of
the pendulum if we would calculate from theory the whole correc¬
tion for reduction to a vacuum.

Since the theoretical solution has been obtained in the case of
a long cylindrical rod, or of such a rod combined with a sphere,
we may regard a pendulum formed in this manner, and which is
convertible in air, as also convertible in vacuum, for it is of small
consequence whether the pendulum be or be not really convert¬
ible in vacuum, provided that if it be not we know the correction
to be applied in consequence.

Note A, Article 65.

Let us apply the general equations (2), (3) to the fluid
surrounding a solid of revolution which turns about its axis,
with either a uniform or a variable motion, supposing the fluid
to have been initially either at rest, or moving in annuli about the
axis of symmetry.

In the first place we may observe, that the fluid will always
move in annuli about the axis of symmetry. For let P be any
point of space, and L any line passing through P, and lying in a
plane drawn through P and through the axis of symmetry; and at
the end of the time t let u' be the velocity at P resolved along L.
Now consider a second case of motion, differing from the first in

\

128 ON THE EFFECT OF THE INTERNAL FRICTION OF FLUIDS

having the angular velocity of the solid and the initial velocity of
the fluid reversed, everything else being the same as before. It
follows from symmetry, that at the end of the time t the velocity
at P resolved along L will be equal to u', since the motion of the
solid and the initial motion of the fluid, which form the data of
the one problem, differ from the corresponding quantities in the
other problem only as regards the distinction between one way
round and the other way round, which has no relation to the
distinction between to and fro in the direction of a line lying
in a plane passing through the axis of rotation. But since all
our equations are linear as regards the velocity, it follows that
in the second problem the velocity will be the same as in the
first, with a contrary sign, and therefore the velocity at P in
the direction of the line L will be equal to —u'. Hence v! ■= — u\
and therefore u= 0, and therefore the whole motion takes place in
annuli about the axis of rotation.

Let the axis of rotation be taken for the axis of 0 ; let co be
the angle which a plane passing through this axis and through
the point P makes with the plane of xy , and let v' be the velocity
at P. Then

u~ —v' sin co, v = v cos co, w = 0,

and all the unknown quantities of the problem are functions of t ,
5 , and '3T, where -sr = + 2/ 2 ). Substituting in equations (2) the

above values of u , v, and w, and after differentiation putting co = 0 ,
as we are at liberty to do, we get

^=0 ^=0

dvr ’ dz ’

fdW dV 1 dv' v' \ _ dv f

^ \dz 2 dzr 2 ^ vt dw tv 2 ) P dt

(179).

The first two of these equations give p = a constant, or rather p = a
function of t, which for the same reason as in Art. 7 we have a
right to suppose to be equal to zero. The third equation combined
with the equations of condition serves to determine v.

Now in the particular case of an oscillating disk, the equation
(179) becomes according to the mode of approximation adopted in
Art. 8

c?V _ dv
^ dz z ^ dt

(ISO),

ON THE MOTION OF PENDULUMS.

129

which in fact is the same as the second of the equations (8). The
solution thus obtained is as we have seen

if = t) .( 181 ),

f denoting a function the form of which there is no need to write
down, which satisfies (180) when written for v'. Now it will be
seen at once that the expression (181) satisfies the exact equation
(179), and therefore the approximate solution obtained by the
method of Art. 8 is in fact exact, except so far as regards the
termination of the disk at its edge, which is what it was required
to prove.

Passing from semi-polar to polar co-ordinates, by putting
z = r cos 0, vr = r sin#, we get from (179), after writing pip for

dW 2 dv' 1 d ( . a dv'\ v _ 1 dv'

dr* + r dr + r* sm9 d9\ m dOJ r*&m*9 pi dt ''

Suppose now the solid to be a sphere, having its centre at the
origin. Let a be its radius, y its angular velocity, and suppose
the fluid initially at rest. Then v f is to be determined from the
general equation (182) and the equations of condition

v' = 0 when t = 0, v = a y sin 9 when r = a, v' = 0 when r = oo .
All these equations are satisfied by supposing

v' — v" sin 9 }

v" being a function of r and t only.

d V 2 dvf ^ 2F
dr 2 r dr r 2

We get from (182)
pf dt

(183).

If we suppose y constant, v" will tend indefinitely to become

constant as t increases indefinitely, and in the limit = 0, whence

dt

we get from (183) and the equations of condition i= when
r = a, v' —0 when r = oo ,

, W
r*

t * n

v = ~ sm 9.
r

This is the solution alluded to in Art. 8 of my paper On the
Theories of the Internal Friction of Fluids in motion , <&c.

s. III.

9

130 ON THE EFFECT OF THE INTERNAL FRICTION OF FLUIDS

Note B, Article 65.

Let ns resume the problem of Art. 7, but instead of the motion
of the plane being periodic, let us suppose that the plane and fluid
are initially at rest, and that the plane is then moved with a con¬
stant velocity V, and let the notation be the same as in Art. 7.

The general equations (8) remain the same as before, but the
equations of condition become in this case

v = 0 when t = 0 from x = 0 to x~cc }
v = V when x = 0 from t = 0 to t = oo .

By Fourier’s theorem and another theorem of the same kind, v
may be expanded between the limits 0 and oo of x in the following
form:

2 r°o /* oo

v = - cos ax cos ax' <£ (x' } t) doc da

7T J o JO

2 r<x> Too

-f - sin ax sin ax'ylr (x f , t ) doc da .(184).

7T Jo Jo

In fact, v could be expanded by means of either of these expres¬
sions separately, and of course can be expanded in an infinite
number of ways by the sum of the two. If however v had been
expanded by means of the first expression alone, its derivatives
with respect to x could not have been obtained by differentiating
under the integral signs, inasmuch as the derivatives of an odd
order do not vanish when x = 0, but would have been given by
certain formulae which I have investigated in a former paper*. A
similar remark applies to the second expansion, in consequence of
the circumstance that v itself and its derivatives of an even order
do not vanish with x. But by combining the two expansions we
may obtain the derivatives of v , up to any order i that we please
to fix on, by merely differentiating under the integral signs. For
we may evidently express the finite function v, and that in an
infinite number of ways, as the sum of two finite functions <fi (x, t) }
y}r(x, t) which like v vanish when x — qo , and which are such that
the odd derivatives of the first, and the even derivatives of the

* On the critical values of the sums of periodic series. Camb. Phil. Trans., Vol.
vm. p. 533. [Ante, Vol. i. p. 287.]

ON THE MOTION OF PENDULUMS.

131

second, up to the order i, as well as 'yjr (x, t ) itself, vanish when
x = 0. Substituting now in the second equation (8) the expression
for v given by (184), we see that the equation is satisfied provided

f + ^=°, 0.

These equations give

(p (%, £) = y i x ') ^ (%', t) = <r (x f ) e _/x ' a2!f ,

where y, a denote two new arbitrary functions. Substituting in
(184), and then passing to the first of the equations of condition,
we get

0=%(tf)+o-O),

whence a (x) = — y ( x ) an( i

2 r&o r oo

v = — I cos a (x + x) e y (as) das doc
7T J 0 Jo

1 foo ( x'-hx ) 2

= V^t I o 6 " 'X'(®) dx ' .( 185 )-

The second of the equations of condition requires that

1 f 00 X>1 2 f 00 —

V = -r== e 4fi't y ( x ) dx = ~r~ I € ~ s2 X (% s ^ A 6 t) ds.

V 7 TjJL tJ 0 V W J 0

Since the second member of this equation must be independent of
t, we get x ( x ) = a constant, and this constant must be equal to
V, since

v 7r J o

Substituting in (185) we get

y r 00 (x+xT

v= —— e Vt cte'.(18G).

For the object of the present investigation nothing is required but
the value of for x = 0, which we may denote by (^~) • We
get from (186)

Wo”"^ . ( ' 187 ' ) ‘

Now suppose the plane to be moved in any manner, so that its
velocity at the end of the time t is equal to /(£). We may
evidently obtain the result for this case by writing/' (t r ) dt' for V,

132 ON THE EFFECT OF THE INTERNAL FRICTION OF FLUIDS

and t — l! for t in (187), and integrating with respect to t\ We
thus get

jdx)

To apply this result to the case of an oscillating disk, let

dd

r ~~~rF(t) he the velocity of any annulus, and G the moment of

CLu

the whole force of the fluid on the disk Then

o-Wi’[/(£)*■■

and will be got from (188) by substituting rF(t) for /(£).
We find thus

= F'it-t,)-.} .(189).

J 0

If we suppose the angular velocity of the disk to be expressed
by A si Tint, where A is constant, we must put F(t) = A sin nt in
(189), and we should then get after integration the same ex¬
pression for G as was obtained in Art. 8 by a much simpler
process. Suppose, however, that previously to the epoch from
which t is measured the disk was at rest, and that the subsequent
angular velocity is expressed by A t sin nt, where A t is a slowly
varying function of t Then

F(t) = 0 when t < 0, F(t) = A t sin nt when t > 0.

On substituting in (189) we get

G = - sjirfj !. pa 4 n f At-t x cos n(t — O .(190).

Jo V*i

Now treating A t as a slowly varying parameter, we get from a
formula given by Mr Airy, and obtained by the method of the
variation of parameters,

dA t G. /-mi\

-^-=ysm nt .(191),

where I denotes the moment of inertia. In the expression for G
we may replace A t _ ti under the integral sign by A t outside it,
because A t is supposed to vary so slowly that does not much
differ from A t while t x is small enough to render the integral of

ON THE MOTION OF PENDULUMS.

133

importance. Making this simplification and substituting in (191)
we get

^L=-csinntf cos n(t — t^) —y~ .(192),

A t dt Jo V t x

where c= . pa^nl~\ If then A 0 be the initial and A the

final value of A t} we get from (192)

log ~A = ^ f j s * n ^J cos ^{t — t^ dt ...(193).

Let now A 0 + A A 0 be what A Q would become if, while the
final arc A and the whole time t remained the same, the motion
had been going on for an indefinite time before the epoch from
which t is measured, in which case the superior limit in the
integral involved in the expression for Q would have been oo in
place of t. Then

log = cj jsin ntj cos n (t — t ± ) efe...(194),

whence by subtracting, member from member, equation (193)
from equation (194), we get

log

A 0 + AAq
Aq

= o j jsin nt J cos n (t —

which becomes after integration by parts

log

A 0 + A^.o
A

= ™ | \J • cos cos ^ J cos

r°° # dt
+ (2 nt — sin 2 nt) / sin nt -j-
J t v&.

dt

Tft

..(195).

Now t is supposed to be very large: in Coulomb’s experiments in
fact 10 oscillations were observed, so that nt = 107T. But when t
is at all large the two integrals

r dt r • * dt

cos nt- 77, I sm nt —

Jt v* Jt yt

can be expressed under the forms

— P sin nt -f Q cos nt, P cos nt + Q sin nt,

P = vT 1 t~^’-l. 3.2~ 2 n~ z t~% +

Q= 1.2' 1 n~ 2 1~% — 1.3.5.2 -3 rT 4 1~ « + ...,

where

134 ON THE EFFECT OF THE INTERNAL FRICTION OF FLUIDS

series which are at first rapidly convergent, and which enable us
to calculate the numerical values of the integrals with extreme
facility. These expressions were first given by M. Cauchy, in the
case of Fresnel’s integrals, to which the integrals just written arc
equivalent. They may readily be obtaihed by integration by
parts, though it is not thus that they were demonstrated by
M. Cauchy. If now the above expressions be substituted for the
integrals in (195) the terms containing t'- destroy each other, and
for general values of t the most important term after the first
contains Since however t is supposed to correspond to the
end of an oscillation, so that nt is a multiple of v, the coefficient
of this term vanishes, and the most important term that actually
remains contains only t~§. Hence neglecting insensible quantities
we get from (195)

log

A a + A.4 0

A

C /tt

4 n V 2>i

.(19C).

We get from (194) by performing the integrations

log ^o .

sin nt (cos nt -f sin nt) (It

=ii \/£ f 2 nt +1 ~ cos 2,it ~ siu 2riV > >

which becomes since nt is a multiple of ir

, A^A-AA^
log . -
° A

We get from (196) and (197)

) = C / W
in V 2 n'

:tut

(197).

2 nt log
whence

A q 4- Azl

ZT

°=l0£f

A {) + LA,
A

„ Z 0 -f- AA „

: A 0

log^^^CS^-D-lo,^’ .(IthS),

and the same relation exists between the common logarithms of
the arcs, which are proportional to the Napierian logarithms.
Now Log J. 0 - Log.4 is the quantity immediately deduced from
experiment, and Log(4 # + Ail,)- Logyl,, is the correction to be
applied, in consequence of the circumstance that the motion began
from rest. Instead of applying the proportionate correction

ON THE MOTION OF PENDULUMS.

135

+ (2nt — l) -1 to the difference of the logarithms, we may apply it
to the deduced value of which is proportional to the difference
of the logarithms. In Coulomb’s experiments 10 oscillations were
observed, and therefore 2 nt = 207r, and (2 nt-~ 1) _1 = 0’01617, and
the uncorrected value of Va 6 ’ being 0*0555, we get 0*0009 for the
correction, giving \ffjS = 0*0564.

Note C. Article 50.

The results mentioned in this article were originally given
without demonstration; but as the mode in which they were
obtained is short, and by no means obvious, I have thought it
advisable to add the demonstrations.

In order that the right-hand members of equations (138) may
be perfect differentials, we must have

dB

da >'" _

= 0,

dB

dco'

0,

d8 +

dco"

= 0.

..(a),

dii +

dx

dz

dy ~

dx

dz

dB_

doo"

= 0,

dB

da>'"_

0 ,

dB

dco'

..(b),

dz

dx

dx

dy ~

dy

dz

= 0 .

d(o"

dco'" _

= 0,

dcor

dco'

0,

dco'

dco"

= 0.

..(c).

dy +

dz

dz

dx

dx

dy

The equations (c) give

dc o __ d(o" ___ ^ dco"

dx~~°’ dy~ 3 IT

(d).

In the particular case in which 3 = 0, the equations (a), (6), and
(d) give

day' = 0, day' = 0, dco'" = 0,

and therefore go', go", and co" are constant as stated in Art. 50.
In the general case the equations (a), (6), and (d) give for the
differentials of go', go" , and go'" the following expressions :

dS , . dS , \

136 ON THE EFFECT OF THE INTERNAL FRICTION OF FLUIDS

In order that the right-hand members of these equations may be
perfect differentials, we must have

_*L.o

dzdx ’

d?S cfS_
dy 1 ^ dz*

d?h =
dz* + da?

d?S

dcc 2+ dif ’

integrating, and then substituting in (138) the resulting expressions
for ©', co", a/", and integrating again, we shall obtain the results
given in Art. 50.

[The possibility of a more general kind of motion than that of
a solid taking place in an elastic fluid without consumption of
energy by internal friction, that is, without its. being converted
into the kinetic energy of heat, depends on the coefficient in the
last term of (1) being /x/3, or rather not greater than /x/3; and
that again on the assumption made in a former paper (VoL I.
p. 87) that in any elementary portion of the fluid a velocity of
dilatation alike in all directions does not affect the hydrostatical
relation between the pressure and density. Although I have
shown (Yol. I, p. 119) that on the admission of a supposition
which Poisson would probably have allowed the two constants in
his equations of motion are reduced to one, and the equations
take the form (1), and although Maxwell obtained the same
equations from his kinetic theory of gases (Philosojrfdcal Trans¬
actions for 1867, p. 81) I have always felt that the correctness of
the value fij 3 for the coefficient of the last term in (1) does not
rest on as firm a basis as the correctness of the equations of
motion of an incompressible fluid, for which the last term docs not
come in at all. If the supposition made above be not admitted,
we must replace the coefficient /x/3 by a different coefficient, which

ON THE MOTION OF PENDULUMS.

137

may be written fij 3 + and m must be positive, as otherwise the
mere alternate expansion and contraction, alike in all directions,
of a fluid, instead of demanding the exertion of work upon it,
would cause it to give out work. But if the positive constant
exists, the coefficient of the squared velocity of dilatation in the
transformed expression for LV in p. 69, instead of being — J/t,
will be —1/4 + ot, and in order that the quantity under the sign of
triple integration may vanish, we must have in addition to the
equations (137) on p. 70 the further equation 8 = 0, and the
conclusion is the same as in the case of an incompressible fluid.]

Additional Note. (See foot-note at p. 77.)

[The fact that notwithstanding the great variety in the forms
of Baily’s pendulums, even when restricted to those to which the
theory of the present paper is applicable, the results of his experi¬
ments manifest such a remarkable agreement with theory, in spite
of the adoption in the calculation of a law as to the relation of (m
to p which we now know to be wrong, paradoxical as that fact
appears at first sight, admits of being easily explained by com¬
bining two considerations, relating the one to the way in which
the experiments were conducted, the other to the character of the
formulae.

In the case of the 41 pendulums mentioned in Baily’s paper as
originally presented to the Royal Society, the high pressure under
which each pendulum was swung was always the atmospheric
pressure, and the low pressure did not much differ from that of
1 inch of mercury. There can be little doubt that the same prac¬
tice was followed as regards the 45 additional pendulums for
which the reduced results only, not the details, are given in the
appendix. Hence the two pressures used would be nearly the
same throughout, and nearly those measured by 30 inches and
1 inch of mercury. The ratio of the densities would be very
nearly the same.

The expression (52) shows that for a sphere k is of the form
k = A + B\J p! .(a),

where A is an abstract number, and B depends on the diameter
and time of vibration of the sphere, but is constant when only the

138 ON THE EFFECT OF THE INTERNAL FRICTION OF FLUIDS

nature and the density of the gas are changed. In the case of a
long cylinder, the expression for k is complicated, and involves
infinite series. Nevertheless the numerical table on p. 52 shows
that if we except the very early part of the table, which corre¬
sponds to very slender rods, or (according to Maxwell’s law) to
very low pressures, the value of k nearly fits an expression of the
form (a), A being in this case 1, instead of which was its value
for a sphere, and B depending, as it did before, on the diameter
and time of vibration. And if the pendulum be made up of a
sphere and a cylindrical rod, as was the case with many of Baily’s,
we shall still not go far wrong if we take lc to be expressed in the
same form (a), in which the constants A and B admit of being
obtained by calculation. Now by the definition of k the effect of
the air on the time of vibration of a given pendulum varies as
(1 + k) p. Baily’s n, or 1 + k, was got by dividing the observed
difference in the time of vibration at the high and low pressure,
corrected for everything but the effect of the air, by the calculated
difference for buoyancy alone.

Hence n = 1 + AkpjAp, where A denotes the difference at the
high and low pressures. If we assume k to be given nearly
enough by the formula (a), we have

n = 1 + A + B .

Ap

If we put BG for u— 1 — A, we have according to the assumed
law *J/jl =C } and therefore if we denote the higher and lower
densities by p v p 0 , we have at the atmospheric density

= CWp l ,

which gives what p is supposed to be. But we ought to have
taken

(J=Jp.

A \!p
Ap ’

so that if we denote the apparent coefficient by 'p, reserving p for
the true coefficient, we have

bp

Vp,A\/p

P, - P» = t +
'/PiWPi- Vpo)

£o

Pi ’

Now in the same experiment the swings tit high and low
pressure were taken at temperatures that did not much differ;

ON THE MOTION OF PENDULUMS.

139

and indeed Baily arranged the order of the swings in such a
manner as to eliminate the effect of a small progressive change.
We may therefore take the ratio of the densities as being that of
the pressures, which, as already stated, was nearly that of 30 to 1.
Hence the values of p, which would be deduced independently
from the different experiments on the supposition that pi and not
p was independent of the density would all be wrong in nearly the
same ratio, which would be nearly that of (\/30 + l) 2 to 30. This
accounts for the remarkable agreement between theory and expe¬
riment as regards the time of vibration, notwithstanding the
employment of an erroneous law as to the relation between p,
and p in making the correction for the residual air at the low
pressure. Moreover, in order to arrive at the value of p which
would have been deduced from the experiments on the time of
vibration had they been reduced according to Maxwell's law, we
have merely to increase the value as obtained in this paper in a
ratio which is nearly that given above, or 1 to 1*398. The mean
high and low pressures for the four 1^-inch spheres were 30*062
and 1*177, numbers which would give for the factor 1*558. Of the
three pendulums in the table on p. 79, from which the adopted
value of \fp! was deduced, the first is the only one for which the
pressures are recorded in Baily’s paper, and the calculated factor
for it is 1*437.

The results of the most recent and trustworthy experiments for
the determination of p for air are brought together by Mr H. Tom¬
linson in a paper published in the Philosophical Transactions for
1886, p. 767. From the numbers given by him on p. 768 (first line
in the table), and on pp. 784, 785, it appears that the true factor
should be about 1*700. It is *Jp that enters into the expression
for the time of vibration, and the difference between the square
roots of 1*7 and 1*4 is only *0925 of the former; and it is only a
portion of the correction for inertia in which the viscosity is
involved at all. Thus in the case of the lj-inch spheres (see
p. 86), that part of the correction for the air in which alone the
viscosity is involved is little more than the one-seventh of the
whole; and a fraction of this again which is barely one-tenth
would amount to little more than one per cent, of the whole effect
of the air. Considering the uncertainties as to some small cor¬
rections, such as that for the effect of the confinement of the

140 ON THE EFFECT OF THE INTERNAL FRICTION OF FLUIDS

space in which the pendulums were swung, the resistance to the
wire in the case of the spheres, the deviation of the motion of the
air near the bottom of long cylindrical rods from a motion in two
dimensions, in a horizontal plane, such small discrepancies as that
just noticed can hardly he affirmed to be real, that is, such as
would emerge from mere casual errors of observation if the above
corrections were made perfectly. It is however possible that with
amplitudes of vibration as large as those actually used, amounting
to about 1° to start with, there may have been a very slight pro¬
duction of eddies, the effect of which on the time of vibration may
not have been wholly insensible.

But it is not only with regard to the time of vibration that the
results of Baily’s experiments manifest such a remarkable agree¬
ment with theory notwithstanding the adoption of an erroneous
assumption as to the relation of p to p ; the observed reduction of
the arc of vibration was also found not greatly to differ from that
given by calculation.

As regards the spheres, the equation (172), p. 114, gives

Now even in the case of the l-?,-inch spheres the right-hand
member of this equation exceeds unity by only 0*127, so that the
supposition that, in order to rectify the law connecting p with p ,
h' requires to be altered in the same ratio as k —does not much
differ from the truth. Accordingly the calculated numbers for the
spheres (p. 120) differ but little from the numbers given by obser¬
vation. Calculation by aid of the table on p. 52 shows as regards
the rod No. 21 that if the calculated number 287 represented the
number given by observation, the assumed value ((HIOf of p
for atmospheric density would have to be increased in the ratio of
1 to 1*400, bringing it considerably nearer to the true value as
given by modern determinations. As observation gave for the
logarithmic decrement 315, p would have to be still further
increased, making it somewhat too high. Similar remarks apply
to the tube Nos. 35—38, which however according to Baily’s figure
differs much from a plain cylinder, and which moreover requires a
comparatively large correction for confined space. On the whole
then the pretty close agreement between theory and observation as
to the decrement of arc, notwithstanding the assumption of a

ON THE MOTION OF PENDULUMS.

141

wrong law in the reduction, may be considered to have been
accounted for.

The very accurate experiments of Bessel are unfortunately not
available for more than a very rude comparison, for the reason
mentioned in the foot-note at p. 97.

As regards the correction on account of the air to the time of
vibration of a pendulum, we have seen that in the case of a sphere,
and very approximately in the case of a long cylindrical rod which
is not extremely narrow, it is of the form

Gp +

where G and H depend on the form of the pendulum, but not
upon the pressure, nor indeed on the nature, of the. surrounding
gas, which might be other than air. There can be little doubt
that the same would apply as a very near approximation to
any of the ordinary forms of pendulum, though in that case the
constants G, H cannot in general be obtained by calculation. The
first term depends partly on buoyancy, partly on the inertia of the
gas regarded as a perfect fluid. As the latter part cannot be
calculated, there is no need to calculate the former, since the two
have to be determined as a whole by observation. As the value
of p for air is now well known, the constants G and H may be
determined from the differences in the times of vibration at three
suitably distributed pressures. These constants are determined
once for all for the same pendulum. They may even be applied
without a fresh experimental determination to any other pendulum
of which the external form is geometrically similar, even though
the internal distribution of mass be different, of course with due
regard to the dimensions of the terms with respect to the units of
length and time. Moreover unless we want to combine observa¬
tions with different gases, or else to take account of the variation
of p with temperature, we may write the above formula

Gp + H' Vp,

and as we must appeal to experiment for the determination of H ',
we do not even need to know the value of p.]

[From the Philosophical Magazine, Yol. I., p. 305 (April, 1851).]

An Examination of the possible Effect of the Eadiation
of Heat on the Propagation of Sound*

Inasmuch as Laplace’s formula is a rigorous deduction from
the physical hypotheses adopted, there is no way of escaping from
his result but by calling in question the hypotheses themselves.
Now the development of heat and cold by sudden condensation
and rarefaction is not merely a hypothetical cause, the only
evidence of whose existence is that it explains the phenomena,
but is a well-known physical fact, proved by direct experiment.
That in the case of small sudden condensations (positive or nega¬
tive) the increase of temperature is ultimately proportional, ctuteris
paribus , to the condensation, will not, it is presumed, be called in
question. The only way, then, of escaping from the conclusion
that the velocity of sound is really increased by the cause assigned
is, to suppose that the heat produced by condensation passes away
so rapidly by radiation that the result is the same as though con¬
densation and rarefaction were incapable of changing the tempe¬
rature of air. The main object of the present communication is
to examine the consequences of such a supposition, in order to
make out whether it be tenable or not.

Let us take the case of an infinite mass of homogeneous elastic
fluid, acted on by no external forces, and having throughout a
uniform temperature, and consequently a uniform pressure, except

* [This examination was made in consequence of the publication in the Philo¬
sophical Magazine of some papers in which the correctness of Laplace ’h explanation
of the excess of the observed velocity of sound over that calculated by Newton was
called in question. In the reprint, a few passages which are merely controversial,
and of ephemeral interest, are omitted.]

THE RADIATION OF HEAT ON THE PROPAGATION OF SOUND. 143

in so far as the pressure, and consequently the temperature, are
affected by small vibratory movements. Let the fluid be referred
to the rectangular axes of x, y, z; let % v, w be the components of
the velocity, t the time, p the pressure, p the density in equi¬
librium, p (1 -b s) the actual density, so that s is the condensation.
The three ordinary equations of motion and the equation of con¬
tinuity become in this case, on neglecting as usual the squares of
small quantities,

dp __ __ du dp _ dv dp _ dw
dx ^ dV dy ^ dt 3 dz ^ dt

ds du dv dw __ n
dt dx dy dz

( 1 ).

( 2 ).

Let 0 O be the temperature in equilibrium, 0 Q + 6 the actual
temperature. Then p = k 0 p (1 + s)(l + ol 0 6 q + dp Putting k for
& o (l + a o 0o), a ^ 0r a o(l + a o^o) -1 j an( l neglecting the product of s
and 6, which are both small quantities of the first order, we get

p = kp (1 + s + ad) .(3).

It remains to form the equation relating to the changes of
temperature. Let {is be the elevation of temperature produced
by a sudden small condensation s. The condensation which a
given element of the fluid receives in the time dt is equal to s dt,
where

, ds ds ds ds

S — ~j7 4* U 7 — b V -j —b W -j
at dx dy dz

ds .

dt> nearly;

and the elevation of temperature due to this condensation is equal
to /3s'dt. We know that heat radiates freely to great distances in
air, and therefore, of the heat which radiates from the element
considered, we may neglect the small portion which may be
absorbed by the air in its neighbourhood, and consider only what
goes to great distances. Hence the result will be sensibly the
same as if the element radiated into a medium having the con¬
stant temperature 0 Q) which is the mean temperature of the whole.
The quantity, then, which escapes from the element during the
time dt, will be proportional to the small excess 6 of the tempera¬
ture of the element over the mean temperature of the medium;
and the consequent depression of temperature may be expressed
by qddt, where q is a constant which may be called the velocity of

144 AN EXAMINATION OF THE POSSIBLE EFFECT OF THE

cooling referred to a
therefore,

difference of temperature unity.

dd n ds a

dt-P s- 3 "-

We have,

.(4)-

The six general equations (1), (2), (3), (4) serve, along with
the equations of condition relating to any particular problem, to
make known the six unknown quantities u, % w, p , s , 8.

To simplify the question as much as possible, I shall take the
case of plane waves. Taking the axis of x perpendicular to the
planes of the waves, we have v = 0, w = 0, and u, p , s, 8 will be
functions of only x and t The equations (1) and (2) become

dp __ du ds du
dx — P dt ’ dt^dx

.(«);

and eliminating p and u from these equations and (3), we get

dM\

df \dx 2 a dx 2 )

( 6 ).

Eliminating 6 between (4) and (6), we get

dh __
df~

(l+*$)j t + q

d?s
dx 2

(7).

This equation is satisfied by

s = A. / e mx + n ' t ,

( 8 ).

where A f is an arbitrary constant, real or imaginary, and n
are two real or imaginary constants connected by the equation

{n' 4 - q)n
h {(1 + a/3) n f -l- q}

(9).

If we suppose m r wholly imaginary, the formulae will refer to an
infinite series of waves, the expressions for s, &c., involving x
under the circular functions sine and cosine. In this case our
formulae would make known the manner in which the motion
alters with the time. If we suppose n' wholly imaginary, the
motion will be periodic as regards the time. In this case we
must not suppose the fluid unlimited, but bounded in one direc¬
tion by a vibrating plane which keeps up the motion. I shall
select the second case for consideration, inasmuch as it is analo¬
gous to that of the vibrations propagated along a long tube from
a sonorous body at one end of it, and accordingly will bear on the

RADIATION OF HEAT ON THE PROPAGATION OF SOUND. 145

experiments by which M. Biot proved that the velocity of propa¬
gation of sound in air is independent of the pitch.

Let the origin be situated at the vibrating plane, and let us
consider the motion of the fluid situated at the side of x positive.
Let m be what m' becomes when n is replaced by J — In. The
equation (9) furnishes two values of m, corresponding to two
series of waves, which travel, one in the positive, and the other
in the negative direction. Of course we are only concerned with
the former. We get from (9)

7Y1 2 = — fJ? (cos 2 y/r — J— 1 sin 2^).(10),

where

n f n 2 + q 2

^ ~ VH(1 + *$fn % + qV . K }

-j (l+ap)n

■ tan” 1 - = tan” 1

a/3nq

.( 12 ).

2llr = tan — — U«/J_L VIALJ. -; /Ci\ 2 i -2

r q q (1 + a!3)n 1 + q i

Choosing that root of m 2 which corresponds to waves travelling
in the positive direction, we get from (10)

m = — J—lfjL (cos y/r — J— 1 sin ^r).

Substituting in (8), introducing another function got by changing
the sign of J— 1 and taking a now arbitrary constant, changing
the arbitrary constants so as to get rid of the imaginary quanti¬
ties, and altering the origin of the time so as to get rid of one of
the circular functions, we get

s = * cos (nt — fju cos yjr . x) .(13).

It will be easily seen that the expressions for 0, u , and p are
of the same form, that is to say, that they involve the same expo¬
nential multiplied by a sine or cosine of the same angle. Had
the actual expressions been required, it would have been shorter
to defer the substitution of real for imaginary quantities until
after the imaginary expressions for 9 , u, and p had been obtained.

Now the formula (13) shows, that unless sin ^ be insensible,
sound cannot be propagated to a distance, but must be stifled in
the neighbourhood of the vibrating body by which it is excited.
Since we know very well that this is not the case, we arc taught
that sin yjr is insensible, and therefore ^ itself, since yjr denotes an
angle lying between 0 and 7t/4. The formula (13) shows, that
if V be the velocity of propagation, V = n/jT 1 sec yjr, which, when

10

s. in.

146 AN EXAMINATION OF THE POSSIBLE EFFECT OF THE

f is insensible, reduces itself to n/uT 1 . Referring to (12), we see
that, in order that ^ may be insensible, it is necessary to suppose,
either that 2 is incomparably greater than n, or that n is incom¬
parably greater than q. On the former supposition the formula
(11) gives V=z\/Ic, which is equivalent to Newton’s result. On
the latter supposition we get V = Jh (1 + a/3), which is equivalent
to Laplace’s result.

The reason why sound would be so rapidly stifled were q and
n comparable with each other, may be easily seen on taking a
common-sense view of the subject. Conceive a mass of air con¬
tained in a cylinder in which an air-tight piston fits, which is
capable of moving without friction, and which has its outer face
exposed to a constant atmospheric pressure; and suppose the air
alternately compressed and rarefied by the motion of the piston.
If the motion take place with extreme slowness, there will be no
sensible change of temperature, and therefore the work done on
the air during compression will be given out again by the air
during expansion, inasmuch as the pressure on the piston will be
the same when the piston is at the same point of the cylinder,
whether it be moving forwards or backwards. Similarly, the work
done in rarefying the air will be given out again by the atmosphere
as the piston returns towards its position of equilibrium, so that
the motion would go on without any permanent consumption of
labouring force. Next, suppose the motion of the piston some¬
what quicker, so that there is a sensible change of temperature
produced by condensation and rarefaction. As the piston moves
forward in condensing the air, the temperature rises, and therefore
the piston has to work against a pressure greater than if there
had been no variation of temperature. By the time the piston
returns, a good portion of the heat developed by compression has
passed off, and therefore the piston is not helped as much in its
backward motion by the pressure of the air in the cylinder as
it had been opposed in its forward motion. Similarly, as the
piston continues its backward motion, rarefying the air, the tem¬
perature falls, the pressure of the air in the cylinder is diminished
more than corresponds merely to the change of density, and there¬
fore the piston is less helped in opposing the atmospheric pressure
than it would have been had the temperature remained constant.
But by the time the piston is returning towards its position of

RADIATION OF HEAT ON THE PROPAGATION OF SOUND. 147

equilibrium, the cold has diminished in consequence of the supply
of heat from the sides of the cylinder, and therefore the force
urging the piston forward, arising, as it does, from the excess of
the external over the internal pressure, is less than that which
opposed the piston in moving from its position of equilibrium.
Hence in this case the motion of the piston could not be kept up
without a continual supply of labouring force. Lastly, suppose
the piston to oscillate with great rapidity, so that there is not
time for any sensible quantity of heat to pass and repass between
the air and the sides of the cylinder. In this case the pressures
would be equal when the piston was at a given point of the
cylinder, whether it were going or returning, and consequently
there would be no permanent consumption of labouring force. I
do not speak of the disturbance of the external air, because I am
not now taking into account the inertia of the air either within or
without the cylinder. The third case, then, is similar to the first,
so far as regards the permanence of the motion; but there is this
difference; that, in consequence of the heat produced by compres¬
sion and the cold produced by rarefaction, the force urging the
piston towards its position of equilibrium, on whichever side of
that position the piston may happen to be, is greater than it
would have been had the temperature remained unaltered.

Now the first case is analogous to that of the sonorous vibra¬
tions of air when the heat and cold produced by sudden condensa¬
tion and rarefaction are supposed to pass away with great rapidity.
For we are evidently concerned only with the relative rates at
which the phase of vibration changes, and the heat causing the
excess of temperature 9 passes away, so that it is perfectly imma¬
terial whether we suppose the change of motion to be very slow,
or the cooling of heated air to be very rapid. The second case is
analogous to that of sound, when we suppose the constants q and
n comparable with each other; and we thus sec how it is, that, on
such a supposition, labouring force would be so rapidly consumed,
and the sound so rapidly stifled. The third case is analogous to
that of sound when we make the usual supposition, that the alter¬
nations of condensation and rarefaction take place with too great
rapidity to allow a given portion of air to acquire or lose any
sensible portion of heat by radiation. The increase in the force of
restitution of the piston, arising from the alternate elevation and

10—2

148 AN EXAMINATION OF THE POSSIBLE EFFECT OF THE

depression of temperature, is analogous to the increase in the
forces of restitution of the particles of air arising from the same
cause, to which corresponds an increase in the velocity of propaga¬
tion of sound.

Another consequence follows from the formula (13), which
deserves to be noticed. We have already seen that this formula
gives n/jT 1 sec ^ for the value of V, the velocity of propagation.
Putting for shortness

1 + a/3 = K .(14),

we get from (11) and (12),

2k(KSi* + f)

V _ Aw 2 + 2 2 + VK^ V + 2 2 )(w 2 + 2 2 )} . K

Hence if q be comparable with n, V, which is a function of the
ratio of q to n, will change with % and therefore the velocity of
propagation will depend upon the pitch, which is contrary to
observation. But if q be either incomparably greater or incom¬
parably smaller than n, V will assume one or other of its limiting
values VMT; and the velocity of propagation will be inde¬
pendent of the pitch, as observation shows it to be. We are thus
led, by considering the velocity of propagation, to the same
conclusion as was deduced from the circumstance that sound is

capable of travelling to a distance.

Since, then, we are driven to one or other of the alternatives
above mentioned, it only remains to decide which wc must choose.
But before entering on this subject, it will be proper to consider
whether the formula (13) is of sufficient generality.

In the first place we may observe, that the formula (13) is
only a particular integral of (7). It is adapted to the case in
which the motion is kept up by a vibrating plane, which agrees
most nearly with the circumstances of ordinary experiments; but
a particular law of disturbance as regards the time is assumed,
namely, that expressed by a single circular function. Now we
know that any periodic function of the time, having r for its
period, may be expressed by the sum of a finite or infinite
number of circular functions having for their periods r and
its submultiples; and even a non-periodic function may be
expressed by a definite integral, of which each element denotes
a circular function. So far, therefore, the formula (13) is of
sufficient generality.

RADIATION OF HEAT ON THE PROPAGATION OF SOUND. 149

In the next place, the formula (13) applies to motion in one
dimension only. Eat had we employed the general equations
(1), (2), which relate to motion in three dimensions, we should
have obtained the same partial differential equation as (7), with
the exception that the last term outside the brackets would have
been replaced by

is_ dh

df + df + dz *'

If now we take the case next in order of simplicity, in which the
motion is symmetrical adbout a centre, and put r for the distance
of any point from the centre, we shall get for the determination
of rs the same partial differential equation as (7), with the ex¬
ception that x will he replaced by r. To obtain, therefore, the
integral corresponding to (13), it will be sufficient to replace x by
r and divide the second member by r. This integral would apply
to the case of the disturbance produced by a vibrating spherical
body, in which the motion is supposed to be symmetrical with
respect to the centre. And in the more general case of a vibrating
body of irregular form, or a musical instrument, or any other
source of sound, the conclusions would doubtless be the same as to
their leading features.

There remains a morn important point to be considered before
we apply the formula (13) to the vibrations of air within a long
tube. At first sight it might seem that the radiation of heat
within a tube must take place in a manner altogether different
from that in which it would take place in free air. But a little
consideration will show, I think, that such is not the case. Of
the heat radiating from any particle of air which has been slightly
heated by condensation, any particular ray is incident on the
side of the tube, where it is partly absorbed, partly reflected,
and, it may bo, partly scattered. The reflected ray, or any one
of the scattered rays, is again incident on the side of the tube,
where a good portion is absorbed, and so on. The small quantity
of radiant heat which remains after three or four reflexions may
be regarded as insensible. Now since radiant heat travels with a
velocity equal to, or at any rate comparable with, that of light, wc
may neglect as altogether insensible the time which any portion
of heat, once become radiant, takes to be absorbed. Moreover,
wc may neglect the small portion of heat reabsorbed by the air

150 AN EXAMINATION OF THE POSSIBLE EFFECT OF THE

itself, because a ray of beat has only to traverse a length of air
comparable with three or four diameters of the tube before it is
absorbed by the tube. Hence we may conceive a small periodic
flux of heat as taking place across the inner surface of the tube.
Now it follows from the mathematical theory of heat, that when
a periodic flux of heat takes place at the surface of a solid, the
corresponding variation in the temperature of the solid near the
surface is very small if the period be very small. If we suppose
the flux expressed by the sine or cosine of an angle proportional
to the time, the expression for the fluctuation of temperature will
involve in its coefficient the square root of the period. In the
present case, the period with which we have to deal is that of a
sonorous vibration, a time which must be regarded as extremely
small in questions relating to the conduction of heat. Hence, if
t be the period of vibration, the fluctuation of temperature of the
tube will be a small quantity of the order V T compared with the
flux of heat. Now if U, h be the interior and exterior conduc¬
tivities, v a normal to the inner surface of the tube, drawn from
the tube inwards, 9' the excess of temperature of the tube above
the mean temperature 0 O ; and if we suppose the surface to be
plane, and to radiate into an infinitely extended medium at a
temperature 9 0 + 6 , where 9 is supposed to be constant as regards
space, but to be a periodic function of the time, we must have at
the inner surface of the tube

rlR'

H c j^+h(9'-9) = 0.

Now, according to what has been already remarked, 9' is a small
quantity of the order V T compared with dffjdv ; and it follows
from the above equation, that d9'jdv is comparable with 9, and
therefore & is a small quantity of the order V T compared with 6 .
Hence, even in the case above supposed, the fluctuation of tem¬
perature of the tube at the surface would be very small. But in
the actual case, the tube radiates, not into an infinite medium,
but merely across the air contained within it, beyond which is
situated the opposite face of the tube, at a temperature equal to
the first face; and therefore the fluctuation of temperature of
the inner surface of the tube will be far smaller than in the
case supposed above, so as to be altogether insensible. Hence
the air radiates within an envelope at a temperature # 0 , so

RADIATION OF HEAT ON THE PROPAGATION OF SOUND. 151

that the radiation takes place as if the air heated by compression
radiated into an infinite medium at a temperature 0 O . Of course
the same reasoning will apply to the apparent radiation of cold.
Hence the formula (13) may be applied without change to the
vibration of the air within a long tube, and accordingly may be
employed in considering the experiments of M. Biot above
alluded to.

The preceding view of the effect of radiation within a tube
is very different from that taken by M. Poisson in his Traite de
Mecanique (vol. ii. art. 665). The latter, however, is contained
in a mere passing remark offered by way of conjecture, and
probably written without much consideration, and therefore ought
hardly to be regarded as supported by Poisson’s authority.

Let us now pass to numerical values, in order to make out,
independently of any assumption respecting the true explanation
of the velocity of sound, whether q must be regarded as very great
or very small compared with n. It follows from (13) that the
decrease of intensity in going one wave’s length in the direction
of propagation is a maximum when tanf, and therefore ^ is a
maximum. Now (12) shows that ^ is a maximum when

% = .(16),

K being the quantity defined by (14). For the above value of q
we get from (11) and (12),

fju = 2yfr — .(17).

The velocity of propagation, which is equal to n/jT l see yfr, does
not much differ from n/f 1 , since as will immediately appear,
is not very large. It may be observed, that the expression for
npr 1 given by the first of equations (17), is a geometric mean
between the velocities of propagation resulting from the theories
of Newton and Laplace.

The value of K , deduced from experiments in which the theory
of sound is not assumed, is about 1*30*, whence 2y[r = 8 n 47'. If

* Poisson, Traite de Mecanique, vol. ii. art. G37. The value deduced from the
observed velocity of sound is somewhat larger, and is more likely to bo correct. I
have employed the value 1*30 in order to avoid arguing in a circle, because I am
reasoning as if the received theory of sound were not established.

152 AN EXAMINATION OF THE POSSIBLE EFFECT OF THE

- I be the index of e in (13) when x is equal to one wave’s length,
we have

fisin^.x^l, /a cos.a? = 2 tt, whence Z = 27r tan i/r, e~ l = (16172;

so that the intensity, supposed to vary as the square of the
amplitude of vibration, would be diminished in the ratio of
2*625 to 1. Supposing the period of vibration to be the ^dth
part of a second, which would correspond to a note of moderate
pitch, and taking the velocity of propagation at 1100 feet per
second, we should have 44 inches for the length of one wave.
Hence in travelling 20 yards, or 16*36 wave-lengths, the intensity
would be diminished in the ratio of (2*625) 16 ' 36 to 1, or about
7 millions to 1. A decrease of intensity like this is utterly
contrary to observation, and therefore we are really compelled
to suppose that the ratio of q to n is either very much greater or
very much less than what has just been determined. Since in
the case supposed n = 27tt~ 1 = 6007t, we get from (16)

2 = 2198.(18),

which, it is to be remembered, is referred to a second as the unit
of time.

Let us now, adopting this value of q , examine a little at what
rate a small portion of heated air, situated in other air which has
not been heated, would cool by radiation. If 9 be the excess of
the temperature of the heated air over that of the surrounding
air, we should have, supposing 6 to be suiliciently small to allow
us to adopt Newton’s law of cooling,

from which it follows that the excess of temperature would be
diminished during the time t in the ratio of e (It to I. It would
follow from the numerical value of q above given, that, even in so
short a time as the hundredth part of a second, the temperature
would be reduced in the ratio of about 3514 millions to 1. Such
rapidity of cooling as this is utterly contrary to observation. Put
a poker into the fire, and when it is hot look along it, and an
ascending stream of heated air will be rendered visible by the
distortion which it produces in objects seen through it, in con¬
sequence of the diminution of refractive power accompanying
the rarefaction produced by heat. But were the rate of cooling

RADIATION OF HEAT ON THE PROPAGATION OF SOUND. 153

anything like what has just been determined, no such stream
could exist. Yet we have seen that the observed fact, that sound
is propagated to a distance, obliges us to suppose that the rate
of cooling is either immensely greater or immensely less than
corresponds to q = 2198. It is needless now to say which alter¬
native we must choose. Accordingly, no doubt whatever exists
as to the correctness of Laplace’s explanation of the excess of the
observed velocity of sound over that calculated by Newton.

Now that it has been decided which of the two ratios n : q and
q:n we must regard as extremely small, we may simplify the
formula (13) by retaining only the first power of the ratio in
question, and we shall thus be the more readily enabled to see
in what direction we must look for the first faint indications of
the effect of radiation. Retaining only the first power of q, and
putting n=Vfju, jj, = where V = *J(kK), the velocity of

propagation, wc get from (11), (12) and (13),

s = Ae~ d-^^cos — (Vt- x) .(19).

Hence it is to a diminution of intensity, rather than to an alter¬
ation of velocity corresponding to an alteration of pitch, that we
are to look for the effect of radiation. Now that the objection
raised against Laplace’s explanation of the velocity of sound has
been answered, we may Lake 1*414 for the value of Ic, this being
the mean of the values (pioted by Poisson in art. 604, which were
deduced from the velocity of sound, and arc probably nearer the
truth than the somewhat smaller values determined by a different
process. Putting K = 1*414, V = 1100, taking the square of the
coefficient as a measure of the intensity, and putting N : 1 for
the ratio in which the intensity is diminished while the sound
travels, without divergence, over a length x, wc get

log 1() iV"= 0*000115 tiqx .(20),

the units of time and space to which q and x are respectively
referred being a second and a foot.

From the account of M. Biot’s experiments given by Sir John
Herschel in art. "24 of his Treatise on Sound*, it would seem that
the diminution of intensity which we can by any possibility refer
to radiation must be very small, especially when we remember
K Kitcijrloptcdia M^lropolitana, art. Sound.

154 THE RADIATION OF HEAT ON THE PROPAGATION OF SOUND.

that, in the case of these experiments, the intensity would he
diminished by a sort of reflexion at the bendings* of the tube,
as well as by the friction of the air against the sides of the tube,
and the internal friction of the air itself. That the cause last
mentioned would produce a small but not utterly insensible effect
in causing a diminution of intensity, I have shown in the course
of a paper “ On the Effect of the Internal Friction of Fluids on
the Motion of Pendulums/' recently read before the Cambridge
Philosophical Society*. If we suppose, at a venture, that a dimi¬
nution of intensity in the ratio of 2 to 1 is the utmost which we
can attribute to radiation in the case of M. Biot's experiments,
putting N=% and a; =3120, the length of the tube in feet, we
get from (20) q = 0*834 for a superior limit of q. If we suppose
q — 0*834, we get for the ratio in which the temperature of a
small portion of slightly heated air would be diminished in the
course of one second, 1 to e~ q , or 1 to 0*4343, or 7 to 3 nearly.
It is curious that it should, theoretically speaking, be possible to
assign a superior limit to the velocity ol cooling of heated air by
observations on sound; but I imagine that the. real value of q is a
good deal smaller than any limit which it would bo practically
possible to assign in this way.

I have supposed, as was already observed, that radiant heat
is capable of traversing great lengths of air before any considerable
portion of if is absorbed. This is especially tin; case with heat
of such high refrangibility as to place it within tin* limits of the
visible spectrum ; whereas heat- of low refrangibility, such as that
which would emanate from slightly heated air, is absorbed more
rapidly. Should the distance to which radiant heat* can proceed
in air before a given fraction of it, such as one-half, is absorbed,
not. be extremely great com pared with tin* length of* a wave of
sound, it may be sim alter a little reflection that tin* general
conclusions arrived at will hr unchanged, though the numerical
details would he somewhat altered. 1 have not. met with any
experiments relating to tIn* absorption of non-luminous heat by
air which could be made a foundation for numerical calculation.

[ i«/., v.

[From the Transactions of the Cambridge Philosophical Society , Yol. ix. p. (147)].

On the Colours of Thick Plx\tes.

[Read May 19, 1851.]

The expression “colours of thick plates/' has been appropriated
to a class of phenomena discovered by Newton, and described by
him in the fourth part of the second book of his Optics. In
New ton’s experiment, the sun’s light was admitted into a darkened
room through a hole in the window-shutter, and allowed to fall
perpendicularly upon a concave mirror, formed of glass quick¬
silvered at the back. A white opaque card pierced with a small
hole being then interposed, at the distance of the centre of curva¬
ture of the mirror, so that the regularly reflected light returned
by the same small hole by which it entered, a set of coloured
rings was seen depicted on the card encompassing the hole. The
existence of these rings was attributed by Newton to the light
scattered on entering the glass, and then regularly reflected and
refracted; and he succeeded in deducing from his theory of fits
the laws of the rings, both as regards the relation between the
diameters of successive rings, the order of the colours, the varia¬
tion of the diameter of a given ring corresponding to a variation
cither in the radius of curvature of the surfaces or in the thickness
of the glass, and even the absolute magnitude of the system
formed under given circumstances. The phenomena which present
themselves when the mirror is inclined a little, so as to throw the
image of the hole to one side, arc very curious, and have been
accurately described by Newton in his tenth and eleventh Obser¬
vations.

In the course of a series of experimental researches on these

156

ON THE COLOURS OF THICK PLATES.

rings, the Duke de Chaulnes* discovered accidentally that their
brilliancy was greatly increased by breathing on the glass. Since
the moisture soon evaporated, in order to procure a permanent
tarnish, he spread over the surface a small quantity of a mixture
of milk and water, which on drying left a degree of dimness very
suitable to the experiments. By substituting for the glass mirror
a metallic speculum, in front of which there was placed a plate
of tarnished mica, it was easy to observe the variation in the
diameter of the rings corresponding to a variation in the
distance of the mica from the speculum. In this form of the
experiment the glass plate was replaced by the plate of air com¬
prised between the mica and the speculum. Rings were also
produced when the tarnished mica was replaced by a screen of
fine muslin. In this case, however, according to the Duke’s state¬
ment, the rings were nearly square, though rounded off a little at
the angles. A set of parallel wires gave merely a bright band
intersected by short bands which were vividly coloured. Even
the blade of a knife produced a similar appearance, weak indeed,
but sufficient to establish the identity of the effect. It is un¬
necessary here to discuss the theoretical views of the Duke de
Chaulnes, since the progress of optical science has since led to a
complete explanation of the formation of the rings.

The colours of thick plates were first explained on the un-
dulatory theory by Dr Youngf, by whom they were attributed to
the interference of two streams of light, of which one is scattered
on entering the glass, and then regularly reflected and refracted,
and the other regularly refracted and reflected, and then scattered
on its return through the first surface. I)r Youngs explanation is
however excessively brief; and he has rather pointed out the
a PP^ ca, ^ on tdm grand and newly-discovered principle of inter¬
ference to the explanation of the phenomenon, than followed the
subject into any of its details. At the same lime, it appears
evident, from an attentive perusal of what he has written, that at
least the broad outlines of the complete explanation were clearly
present to his mind.

In the course of a paper entitled u Experiments for investigating
the cause of the coloured concentric rings, discoveied by Sir Isaac.

* Memo ires de V Academia, 1755, p. I'M).

t On the Theory oj Light and Colours. Philosophical Transactions for is02, p. H.

ON THE COLOUES OF THICK PLATES.

157

Newton, between two object-glasses laid upon one another,” Sir
William Herschel mentions an experiment in which rings of the
nature of those of thick plates were produced by merely strewing
hair-powder in the air in front of a metallic speculum placed as
the mirror in Newton’s experiment*. The result of this experi¬
ment was justly regarded by Herschel as inexplicable on the
theory of fits. It may here be remarked that it is in perfect
accordance with the theory of undulations.

In the Annates de Ghimie et de Physique j-, will be found a
report by Ampere and Poisson on a memoir by M. Pouillet, con¬
taining some experiments on the rings. The experiments were
mostly the same as those of the Duke de Chaulnes, but accom¬
panied by measures. M. Pouillet found that the rings were
produced by placing in front of a metallic speculum an opaque
screen containing an aperture of any form. In this case the rings
were round, whatever might be the form of the aperture. The
experiments are mentioned by M. Pouillet in his Elemens de
Physique\.

A complete explanation of the rings, according to the theory of
undulations, has been given by Sir John Herschel in his treatise
on Light§. The rings are supposed to be formed in Newton’s
manner with a glass mirror, the luminous point being situated in
the axis. Having investigated the elementary system of rings
which would be produced by the two streams scattered in passing
and repassing at the point of the first surface where it is cut by
the axis, Sir John Herschel shews that if the surfaces be supposed
to be a pair of concentric spheres, having the luminous point in
their centre, the elementary systems corresponding to the several
elements of the first surface will be superposed, in such a manner
that a distinct system would be thrown on a screen held at the
distance of the luminous point. The laws of the rings resulting
from theory are precisely those which had been discovered by
Newton, and the calculated magnitudes were found to agree
almost exactly with Newtons measures.

A set of coloured bands have since been observed by Dr
Whcwell, which are formed when the image of a candle held near

* Philosophical Transactions for 1807, p. 281.

f Tom. i. (181G) p. 87. X Tom. n. p. 47G.

4$ Encyclopedia Mctropolitana , Arts. 070, &c.

158

ON THE COLOUBS OF THICK PLATES.

the eye is viewed by reflexion in a plane mirror of quicksilvered
glass, placed at the distance of some feet. This observation was
communicated to M. Quetelet, by whom it has been published*.
In repeating the experiment together, Dr Whewell and M.
Quetelet found that it was an essential condition of success that
the surface should not be perfectly bright, and that to ensure the
production of the bands it was sufficient to breathe gently on the
surface of a cool mirror. Instead of vapour, which soon evaporates,
M. Quetelet recommends a tarnish of grease f.

In closing this sketch of the history of the subject, I may be
allowed to express my obligations to Dr Lloyd for his valuable
Report on Physical Optics, which contains a brief account of all
that was known about the subject, accompanied by references to
the original papers.

My attention was called to the subject by the Master of Trinity
College, who shewed me the bands above-mentioned, which lie
shortly afterwards brought before the notice of this Society:];. It
seemed to me from the first that these bands were of the nature
of the coloured rings of thick plates, so that the theory of the
former only required to be worked out, that of the latter being
known. Had I felt any doubt on the subject., if would soon have
been dissipated when I came to make experiments; for by properly
varying the experiments the two systems were seen to be incon¬
testably of the same nature.

The mirrors, whether plane or curved, were prepared in the
following manner, which I can recommend to any one; who wishes
to repeat the experiments, as being both easy and efficacious. The
mirror being held horizontally, a mixture consisting ( ,f three or
four parts of water to one of milk was poured on if, and allowed to
spread over the surface. The mirror was then held in a vertical
position in front of a fire, when the greater part, of the mixture
ran off, and the remainder dried in two or three minutes, when
the mirror was ready for use. To prevent disappointment on the
part of any one who may be looking for Dr Whewells bands, I
will here mention that in order to sec them properly the image of

* Coireapondance Mathv.matiquc vt Physique, Tom. v. p. .T.M.

+ Tom. vi. p. G9.

+ S° e the Philosophical Magazine for April, 1851, p. 3,'hJ.

ON THE COLOURS OF THICK PLATES.

159

the flame must be seen distinctly, so that a short-sighted person
requires an eye-glass or spectacles.

A concave mirror prepared with milk and water is well adapted
for performing Newton’s experiment in his manner, or rather by
substituting, as in the Duke de Chaulnes’s experiments, the image
of the sun in the focus of a convex lens for the small hole em¬
ployed by Newton. The experiment may however be varied in
the following manner. Whatever appearance is presented on a
screen may be seen without a screen by receiving the rays directly
into the eye, and adapting it for distinct vision of an object at the
distance of the screen. Accordingly, in order to see the rings
which in Newton’s experiment were thrown on a screen, it is
sufficient to place a small flame in front of the mirror, in such a
position as to coincide with its inverted image, when a remarkably
beautiful system of rings is seen in air, surrounding the flame.
Not the least striking circumstance connected with these rings is
their apparent corporeity, since they seem to have a definite
position in space like an actual object. The striking and beautiful
phenomena so accurately described by Newton in his tenth and
eleventh Observations may be seen in this manner by moving the
flame sideways. By altering in various ways the positions of the
flame and of the eye, both in this experiment and in that with a
plane mirror, the rings or bands seen in the two cases may be
perceived, independently of any theory, to be evidently of the
same nature. It is unnecessary here to describe at length the
various appearances presented, since they are noticed in the body
of the paper, in connexion with the theory.

The first section contains the theory of the rings formed in
Newton’s manner. The investigation, though differing a little in
the mode in which it is conducted, is the same in principle as
that given by Sir John Herschel, but is somewhat more general,
inasmuch as the curvatures of the two surfaces are supposed to be
any whatsoever, and the luminous point is not supposed to be
situated in the axis. The distance, too, of this point from the
axis is at first supposed to be arbitrary, in order to investigate
under what circumstances the rings can be formed most distinctly
on a screen. The second section contains the theory of the bands
and rings formed by a plane mirror. The expression for the
retardation is deduced as a particular case from the formula

160

ON THE COLOURS OF THICK PLATES.

investigated in the first section; but on account of the interest
which attaches itself to these bands, and the simplicity of their
theory, a separate investigation is likewise given. The next two
sections are devoted to cases of more generality, and on the whole
less interest: still, a few results of some interest are obtained.
The last two sections contain a closer examination of the precise
mode in which the phenomena are produced.

Although the present paper is a little long, the reader must
not suppose that the theory of the rings and bands is anything but
simple. The length arises partly from the detail in which the
subject has been considered, partly from the generality of some of
the investigations, partly from the description of experiments
which accompanies the theoretical investigations.

Section I.

Rings thrown on a screen by a concave mirror consisting of a ;
lens dimmed at the first surface, and quicksilvered at the hack
Condition of distinctness when the rings are thrown on a screen,
or of fixity when they are viewed in air. I nvcstigutlon of the
phenomena observed when the luminous point is moved in a, direction
perpendicular to the axis of the mirror.

1. Let a luminous point L be situated either in or not far out
of the axis of a mirror such as that just described ; and let it be
required to investigate the illumination, at the point M of a
screen, due to two streams of light, of which one is scattered at
the first surface, and then regularly reflected and refracted, and
the other is regularly refracted and reflected, and then scattered
in coming out, the point M being supposed fo be situated not far
out of the axis. Let the mirror be referred fo the rectangular
axes of x, y, z, the axis of z being the axis of the mirror, and the
origin being situated in the first or dimmed surface. Let r be the
radius of the first surface, 5 that of the second, t the thickness of
the glass, g its index of refraction; and suppose r ami .v positive
when the concavities of both surfaces arc turned in the direction of
z positive. Lot a , b, c be the co-ordinates of L ; (f }f (/ those of
M y and suppose a, b, a, and b small compared with r, c, v, and s.
Let x, y, z be the co-ordinates of any point P on the dimmed

ON THE COLOURS OF THICK PLATES.

161

surface, R the retardation of the stream which was scattered at P
on emergence, relatively to that which was scattered at the same
point on entrance. Let L lt L 2 , L 3 be the images of L after
refraction, reflexion, and second refraction, respectively; M 1} M 2 , M Qj
the images of M; and let a , b, c, or a\ b\ c\ with the suffixes
1, 2, 3, denote the co-ordinates of L v Z 2 , Z 3 , or M v M 2 , M s . In
approximating to the value of R, let the squares of the small
quantities, a , b, x , y , &c. be retained, so that the terms neglected
are of the fourth order, since all the terms are of even orders, as
will be immediately seen when the approximation is commenced.

2. The rays diverging from L may after refraction be supposed
to diverge from L 1} notwithstanding the spherical aberration of
direct pencils, and the astigmatism of oblique pencils. For, first,
let L be in the axis. The supposition that the rays diverge from
L 1 is equivalent to supposing that the front of a wave is a sphere
having L 1 for centre, whereas it is really a surface of revolution
such that L x is the centre of curvature of a section made by a
plane through the axis. This plane cuts the sphere above men¬
tioned in a circle, which, being a circle of curvature, cannot have
with the curve a contact lower than one of the second order. But
the contact is actually of the third order, since the curve and circle
touch without cutting. Hence the error produced in the calcula¬
tion of R by supposing the front of a wave to be a sphere, instead
of that surface which it actually is, is only a small quantity of the
fourth order, and quantities of this order arc supposed to be
neglected.

Next, consider an oblique pencil. Let L' and L" be two
points in the axis of the pencil which arc the centres of curvature
of its principal sections. If the distance of L' and L" from each
other, and from L 19 were not small, the front of the wave would
have a contact of the first order with a sphere described round L lt
with such a radius as to pass through the point where the front is
cut by the axis of the pencil; and in that case the error committed
by taking the sphere for the actual front would be of the second
order. But L and L" are situated at distances from L t which are
small quantities of the second order, whence it will readily be
seen that the actual error is only of the fourth order.

3. Let the expression (L to Z 3 ) denote the retardation of a
wave proceeding from L to or rather, in case be a virtual

11

s. III.

JOL OX THE COLOURS OF THICK PLATES.

f'ucus, the dith-ivnee of retardations of two waves starting from L
and 4 and reaching (he same given point. Then

li ~ (/, to 4) + PM - PL S - {(M to IQ + PL - PM a } = K + V,

wilt‘Ft*

K^iLtn L n ) - (.1/ to M a ),

I — PM /V, a — ( PL — PM <t ) =V'~ V", suppose.

According to the* explanation given in the preceding article,
wIhii tin* position of P changes K remains constant, to the degree
of approximation which it is proposed to employ, but the value of
V depends upon the position of P. We have

PM = \ {(i — ./•)" + ( 7 / - yf + (c — z)'\,
ri ‘* \ i 3 —(r : 3 —rr)*;,

, !

an d h > /( t-c I //’ K nearly. Expanding, we get

l r - 2/ 7*'N // ) ! ^ (r/'-.r) # + (&' - yf\ - c a

! — (./■" 4- if) - ^ !(" :i - 4- (A,.y) 2 J.

| > r r * * 1 *7’ where [ r x / is tin* sum of the terms
enitfanuii o olid j * and I v tin* sum ni tdit * terms containin <r b and
//. Th.-n

r , (a

2/ ^ •

Tie-m : no ...v.i i,,n f«> write down since it, maybe deduced
h*«» ! n I 14 v. ii’iici A. // lor r/, ./*. 'Faking D to denote for V wdiat
^ : I * wo got h\ interchanging a and a\ a and a '

and ui.Ma. MC4 ’ 1 :{

( 1 ).

fia* rin-j." may he formed on a screen with perfect
* !l 'oc any that the difference of phase of fho

' *• : r*-ams which ennie from the several points of {lie

* 51 ■' n-nild he the same; in of her words, t had. the
L n>aiId he independent; of ./• an<l y. Deterring for
*ho m >*Migati(»t 1 ot the eonditions of distinctness, we
’La v. hen these conditions are satisfied the expression

ON THE COLOURS OP THICK PLATES.

163

for R must be the same as if % and y were each equal to zero.
We have therefore

R = [L , M]-[M, L],

where

[A M]-(l to +

and [M, L] is formed from [L, M] by interchanging the co¬
ordinates of L and if. In the above expression e 2 has been
written for shortness, in place of cf+b*. Now supposing c, c,, c 2 , c 3 ,
to be all positive, and denoting by A, B the points in which the
first and second surfaces respectively are cut by the axis of the
mirror, we have

(L to Z s ) = AL — y,AL x + yBL 1 + fiBL 2 — y.AL s + AL Z ) ...(2)
which gives, on expanding,

(L to L ,,) — c + ^ — fic t ( c i + *) + 2 (c~+ + ^

ne‘

+ C;

K* _ M _f< +c + e l

v f 10 * 2c + Cs +

2 ( C !i + 0

2c 3

_K e . 2

c+q, + 2 Mt + ^+ 2 a - 2c t ( c 1 + t) 2c,(c, + «)-

Wc have therefore

7? = ~ J -- i ^2 ___ _ _?1__^2_{ ZO\

2 (c/ (c/ + t) c' {o' + t) c x (c x + t) c 2 (c 2 + Oj’ ^

Although this formula was obtained on the supposition that
the points L, L x , Z 2 , X 3 , M, M x , ilf 2 , Jf 3 , lay on the positive side
of the plane of xy, it is true independently of that restriction.
For when one of the foci L, L x , Z 2 , Z 3 , from having been real
becomes virtual, or from having been virtual becomes real, the
corresponding ordinate c, c x , c 2 -f 1, or c 3 changes sign. At the same
time, in the expression for the retardation distance passed over is
converted into distance saved, and vice versa. Hence in any such
expression as (2) the sign of one or more of the lines is changed.
But in the expansion of the radical by which the length of such
line is expressed, the sign of c, c x> c 2 + t, or c 3 must be changed at
the same time, and therefore no change is required in the expanded
expressions.

11—2

164

ON THE COLOURS OF THICK PLATES.

To eliminate e t and e 2 from (3), we may observe that we have
very nearly

e fju c ’ e t c t + 1’

and similar expressions hold good for e' } &c. Hence
e'H c' A. . c/ c 0 ' + A

JB =

2/^c' 2 c x +1

1 +

< +$ c;

+ ,,

2/^c 2 Cj +1 \ o t + t c 2 J

et

c 2 "h A

.(4).

We might, if required, express c x and c 2 in terms of c by the
formulae of common optics, without making any supposition as to
the magnitude of t. In practice, however, t is usually small
compared with c, c jy &c., so that we may simplify the above
expression by retaining only the first power of t. We thus get

2? =

t_(e__i

fM [C* (

l f&/2 _^ + A 2,

fi V o ' 2 c z 4

(5).

4. Before proceeding to apply this expression, let us in¬
vestigate the conditions of distinctness. Denoting by A Ji, i\ v R
the additions to R on account of the terms involving x y y, we get
from (1)

and A y R may be obtained by interchanging a and 6, x and y.
We have by the formulae of common optics

/* 1 ■ 1 1 = 2 1 m

c t r c J c 2 + t s c 1 + i i c 2 r ^

whence, supposing t small, expanding as far as the first power of t y
and putting for shortness

P M-l
s r

(»x

so that p is the radius of a speculum having the same focal length
as the mirror, we obtain

c s P P* l P pi

Qt-i ) 2

‘Z

ON THE COLOURS OF THICK PLATES.

165

and therefore

1/1 _</l __lWl 1

2 \c' c.' c cj n Vc' c ) U' + c

Again

a c, c 2 + t

• - - , a= — a, 2 —-,
flC 2 1 c, +1

whence

C C„

Neglecting t altogether in the formulae (7), we get
l l 2/1 h

and therefore

P> \o p

o

a a,

(9).

.< 10) -

Let A X R 4- A V R = AR. The formulae (G), (9), (10), and the
corresponding formulae relating to A y R, give

The condition of distinctness, as has been already observed, is
that All shall vanish independently of x and y, in which case the
elementary systems of rings corresponding to the several elements
of the dimmed surface will be superimposed on the screen. The
coefficient of x 1 + y 2 will vanish when either of the following
equations is satisfied :

, 112 /lox

o = c, or - .(12).

c c p

In order that the coefficients of x and y may vanish independently
of particular values of a and b' we must have

c' = c = p .(13),

which equations satisfy at the same time both of the equations
(12), of which it would have been enough that one should have
been satisfied. Hence the rings arc formed most distinctly when
the luminous point and the screen arc both at a distance from the

166

ON THE COLOURS OF THICK PLATES.

mirror equal to that of coincident conjugate foci. This agrees
with observation*.

Whatever be the position of the luminous point, if a', b\ c' be
the co-ordinates of its image, we have

a _ a b' _ b 4
c ~ c 3 c'~ c’

and the second line in the expression for A R becomes

2 tfl 1 _ 2 '

flC \c' C p ,

(ax + by),

which vanishes since d satisfies the second of equations (12). If a
screen be held in a direction perpendicular to the axis of the
mirror, at such a distance as to receive a distinct image of the
luminous point, and if aV , c' be now taken to denote the co¬
ordinates, not of the image itself, but of a point of the screen very
near the image, the part of A R which involves the squares of x
and y will continue to vanish, inasmuch as c remains the same as
before, and the part which contains first powers, though not
absolutely evanescent, will be very small; and therefore a portion
of the system of rings in the neighbourhood of the image will be
formed distinctly.

5. This agrees with observation. In repeating Newton's ex¬
periment in his way, except that a lens of short focus was employed
instead of a small hole, and that the surface of the glass was
purposely dimmed with milk and water, I found that when the
mirror was placed at a distance from the luminous point widely
different from its radius of curvature, and inclined a little, so as to
allow of receiving the image on a sheet of paper without stopping
the incident light, and when the paper was held at such a distance
from the mirror as to receive a distinct image of the luminous
point, the image was accompanied by very distinct arcs of rings.

* See Newton’s Optics , Book II., Part iv., Obs. 1, for the case in which the
curvatures of the two surfaces are alike, and an experiment by the Duke de
Chaulnes (Mem. de VAcademic, 1755, p. 141) for the case in which they are unlike.
In this experiment a plano-convex lens was employed. Each face in succession,
after having been tarnished, was turned towards the incident light. It appears
from a passage at the end of Newton’s twelfth Observation that he had himself
made experiments of a similar nature, the results of which however are not
described.

ON THE COLOURS OF THICK PLATES.

167

Whatever appearance is presented on a screen may be seen
without a? screen, by placing the eye in such a position as to
receive the rays, and adapting it to distinct vision of an object at
the distance of the screen in its former position. It is found
universally that when the image of the luminous point is seen
distinctly it is accompanied by a portion, more or less extensive,
of a system of coloured rings or bands. In this way the rings may
be seen when the image is virtual, in which case they cannot, of
course, be thrown on a screen.

In the experiment described in the introduction, in which a
small flame is placed in such a position as to coincide with its
inverted image, and viewed directly, the rings seen are remarkable
for their fixity, appearing like a bodily object surrounding the
flame, and having a definite parallax, whether judged of by the
motion of the head, or by the convergence of the axes of the two
eyes. The same is true of the system of rings formed when the
flame is moved sideways out of the position above mentioned.
The reason of this fixity is, that inasmuch as the retardation is
independent of x and y, a given point of an imaginary plane drawn
through the flame perpendicular to the axis of the mirror belongs
to a ring of the same order, whatever be the point of the mirror
against which it is seen projected.

6. Having investigated the conditions of distinctness, let us
now proceed to consider the magnitude and character of the rings,
supposing the luminous point to be situated at a distance p from
the mirror, and the rings to be thrown on a screen at the same
distance, or else viewed in air. In this case c = c' = p; and if the
luminous point be in the axis e = 0, which reduces (5) to

te'l

/jbd

(14).

It readily follows from this expression that a system of rings is
formed similar to the transmitted rings of the system to which
Newtons name is especially attached. The rings in the present
case, however, especially when viewed in air, arc far more brilliant,
and in this respect more resemble the reflected system. If e t be
the radius of the first bright ring, for which Ii = X, the length of a
wave of light,

c;

e,

168

ON THE COLOUKS OF THICK PLATES.

and for the bright ring of the order n, e f — The formula

(14) has already been discussed by Sir John Herschel,*and com¬
pared with Newton’s measures, with which it manifests a very
close agreement.

With white light, only a moderate number of rings can be
seen, on account of the variation of the scale of the system
depending on a variation in the refrangibility of the component
parts of which white light is made up. When the rings were
formed in air, and the source of light was the flame of an oil-lamp
with a small wick, I have counted seven or eight surrounding the
central bright spot. But when the system is viewed through a
prism, or when the flame of a spirit-lamp is used, an immense
number of rings may be seen.

7. Next, suppose the luminous point out of the axis. Re¬
ferring to the formula (5), we see that the retardation is not now
equal to zero at the axis, but throughout a circle whose radius e'
is equal to e. Hence the achromatic line* of the system, which
was formerly reduced to a point, is now a circle having its centre
in the axis, and passing through the luminous point and its image,
which are situated at the opposite extremities of a diameter. The
fringes of the first order will bo a pair of circles having their
centre in the axis, and lying, one outside, and the other inside the
central fringe: the fringes of the second order will be another
pair of circles lying, one outside the larger, and the other inside
the smaller fringe of the first order, and so on. It is to be
remarked, however, that only a finite number of fringes are formed
inside the central white fringe. If the value of R when e' = 0 be
denoted by - n 0 \, n Q will be a numerical quantity, a function of \
which determines the number of fringes and the fraction of a
fringe, belonging to the light of which the wave-length is A,
which are formed inside the central white fringe. The value of 7 i 0
may be got from equation (5) on putting e' = 0, which gives

te 2
/ jXc

If white light be used, and if n 0 exceed 8 or thereabouts for rays of

* * use tb.is term to denote the locus of the points for which the retardation is
equal to zero, which forms a curve on either side of which the colours are arranged
in descending order.

ON THE COLOUKS OF THICK PLATES.

1C9

mean refrangibiiity, all the fringes which the overlapping of the
different colours allows to be visible are formed inside as well
as outside the central white fringe; and if the luminous point be
moved still further from the axis, a portion of the field of view
around the axis will appear free from rings. If the radius of
the central white fringe, or, which is the same, the distance of
the luminous point from the axis, be denoted by *Jn 09 the radii
of the bright fringes of the first, second, ... orders will be denoted,
on the same scale, by ^(n 0 ±l), *J(n 0 ±2) ... and those of the
dark rings by *J(n 0 ± ■£), \/(n 0 ± f) ...

The manner in which the rings open out from the centre as
the luminous point is moved sideways out of the axis is very
striking, and has been accurately described by Newton. The
explanation of it is obvious. It may be remarked that the system
of rings, regarded as indefinite, is formed on the same scale what¬
ever be the distance of the luminous point from the axis, but the
portion of the indefinite system which alone is visible, in con¬
sequence of the coincidence or approximate coincidence of the
maxima and minima of intensity corresponding to the several
colours, depends altogether upon that distance. Since in passing
from the interior to the exterior boundary of a given fringe the
square of the radius receives a given increase, it follows that the
area of the fringe is constant, that is, independent of the per¬
pendicular distance of the luminous point from the axis. Hence
the breadth of the fringe continually decreases as the diameter of
the circle which forms cither boundary increases. When a small
flame is used for the source of light, and is moved sideways from
the axis, the fringes soon become confused, because a flame which
does very well for forming the broad fringes of comparatively
small radius seen near the axis, will not answer for the fine fringes
of large radius which are formed at a distance from the axis. But
on using for the source of light the image of the sun in the focus
of a small concave mirror belonging to a microscope apparatus, I
found that the fringes were formed quite distinctly even when
their diameters became very large and consequently their breadths
very small.

170

ON THE COLOURS OF THICK PLATES.

Section II.

Bands formed by a plane mirror , and viewed directly by the eye.

8. In the case of a plane mirror p = oo; and if R be the
retardation of the stream scattered at emergence relatively to
the stream scattered at entrance, R will be obtained by adding
together the second members of equations (5) and (11). Hence
we have

1 { (a! _ a j + (y _ 67} - 1 {(* - ay +(y- by}

(15).

It is to be remembered that in this formula a, b , c denote the
co-ordinates of the luminous point; x, y those of any point in the
dimmed surface; a, b' y c' those of any point M of space towards
which the eye is directed, and for distinct vision of which it is
adapted; and that the formula is only approximate, the ap¬
proximation depending both upon the smallness of the obliquity,
and upon the smallness of the thickness t of the glass in com¬
parison with the distances of the luminous point and the point M
from the mirror.

As regards the illumination at a given point M, we are evidently
concerned with so much only of the dimmed surface as lies within
a cone having M for vertex and the pupil of the eye for base; and
the bands will be seen distinctly if R do not alter by more than a
small fraction of X, when x y y alter from one point to another of
the portion of the dimmed surface which lies within this cone.
Now we have seen already that the bands are in all cases seen
distinctly in the neighbourhood of the image when the image
itself is seen distinctly, so that when the image is real the bands
may even be thrown on a screen, in which case a comparatively
large portion of the dimmed surface is concerned in their forma¬
tion. We may conclude that in the present case the bands will
be seen with sufficient distinctness throughout, provided the image
of the luminous point be seen distinctly.

9. In considering the distinctness or indistinctness of the
bands, we are concerned with the finite size of the pupil of the
eye; but in investigating only their form and magnitude we may

ON THE COLOUIIS OF THICK PLATES.

171

supp°s e the pupil a point, and reduce each pencil entering it to
a single ray, which forms the axis of the pencil. Let E be the eye,
or rather the centre of the pupil, h its perpendicular distance from
the mirror, and suppose the axis of £ to pass through E. The ray
by which a portion of a band is seen as if at M cuts the mirror in
a point whose co-ordinates x, y are equal to a', V altered in the
ratio of h to h — c, so that

Substituting in (15), we get
t

P = -

■ay+ (3/ -by}

•(ic),

from which it may be observed that d has disappeared, as evidently
ought to be the case. The expression (16) might have been at
once deduced from (15) by putting the co-ordinates of the eye in
place of a , b\ d. The reason of this is evident, because the
retardation is constant for the same ray, and a ray may be defined
by the positions of any two points through which it passes. We
may therefore employ the points E and P, instead of M and P, to
define the ray, and may therefore at once substitute the co¬
ordinates of E for those of M in the expression for the retardation.

10. To determine the forms, &c. of the bands, nothing more
will be requisite than to discuss the formula (16). As however
this formula was obtained as a particular case from a very general,
and consequently rather complicated investigation, in which the
curvatures of the surfaces were supposed to have any values, and
as the bands to which it relates arc of great interest, the reader
may be pleased to sec a special investigation of the formula for
the case of a plane mirror.

Retaining the same notation as before, except where the con¬
trary is specified, let Z 0 , E lt be the feet of the perpendiculars let
fall from Z, E on the plane of the dimmed surface, and let
L 0 P = s, E 0 P — u. Let Ji t be the retardation of a ray regularly
refracted and reflected, scattered at emergence at P, and so
reaching E\ li 2 the retardation of a ray reaching E after having
been scattered at P on entering into the glass, and let R x — P 2 = R.
Let LSTPE be the course of the first ray, which emanates from

172

ON THE COLOURS OF THICK PLATES.

L, is regularly refracted at 8, regularly reflected at T, and
scattered on emergence at P. The lines LS , ST, TP will evidently
lie in the plane LL 0 P. Let <£, <f>' be the angles of incidence and
refraction at S. We have

R t =LS + ^ST+PE y

= c sec <jf> + 2 /it sec <£' -f *J(h 2 4- u 2 ) .(17),

and

c tan <f) + 2t tan <£' = s, sin <£ = /j, sin <f> .(18).

The obliquity being supposed small, we may expand, and
retain only the squares of small quantities, the terms thus neglected
involving only fourth and higher powers. We get in the first
place from (17)

jR, = c + 2/*« + h + } + 2» + £).(19).

But equations (18) give

and substituting in (19) we get

Tt x = c -f* Qfjut 4" A 4"

flS 2

2 ()C6C 4~ 2£)

/f 2

2A‘

To obtain P 2 , it will be sufficient to interchange c and A, 6* and
u, since if we supposed the course of the ray reversed it would
emanate from E, be regularly refracted and reflected, then
scattered on emergence at P, and so would reach L. Inter¬
changing, subtracting, and reducing, we obtain

R = t {h (fih +2 t)~c(fxc+ 27 )} . (20) '

This formula is more general than (16), since no approximation
has yet been made depending on the magnitude of t. In practice,
however, t is actually small compared with c and h, so that we may
simplify the formula by retaining only the first power of t, which
reduces (20) to (16), inasmuch as

u 2 = x 2 4- y 2 , and s 2 = (x — a) 2 4- (y - bf.

11. Let us now proceed to apply the formula (16) to the
explanation of the phenomena. In discussing this formula, it is
to be remembered that x, y are the co-ordinates of the point of
the mirror on which a fringe is seen projected. Since the direction

ON THE COLOURS OF THICK PLATES.

173

of the axis of y is disposable, we may make the plane of y, z pass
through the luminous point, in which case b = 0, and

/?-?') (a;i+yV) +

2ax a?
c~ c\

...( 21 ).

For a given fringe li is constant. Hence the fringes form a
system of concentric circles, the centre of the system lying in the
axis of x. If a be the abscissa of the centre

ah? __ x / ah ah

?~?“ 5i? U + c + A-<

( 22 ).

Now ah (h — c)” 1 and ah (h + c)™ 1 are the abscissie of the points in
which the plane of the mirror is cut by two lines drawn through
the eye, one to the luminous point, and the other to its image.
Hence we have the following construction : join the eye with the
luminous point and with its image, and produce the former line
to meet the mirror; the middle point of the line joining tint two
points in which the mirror is cut by the two lines drawn from the
eye will be the centre of the system.

Hence if the luminous point be placed to the right of the
perpendicular let fall from the eye on the plane of the mirror, and
between the mirror and the eye, the concavity of the fringes will
be turned to the right. If the luminous point, lying still on the
right, be now drawn backwards, so as to come beside the eye, and
ultimately fall behind it, the curvature will decrease till the
fringes become straight, after which it will increase in the con¬
trary direction, the convexity being now turned towards the right.
This agrees with observation.

12. The expression for li shews that the circle which forms
the achromatic line of the system passes through the two points
mentioned in the last paragraph but one. This is always observed
to be true in experiment as far as regards the image, and is (bund
to be true of the luminous point also when it is in front of the eye,
so as to be seen along with the fringes, provided the fringes reach
so far.

Denoting by n {) X the value of li at the centre of the system of
circles, taken positively, we get from (21) and (22)

__

°~M^ 2 ~cr)

n.

( 2 »).

174

ON THE COLOURS OF THICK PLATES.

The numerical quantity n 0 may conveniently be called the central
order , since when it lies between i — ^ and % + where % is any
integer, the colour at the centre belongs to the bright 'ring of the
i th order. If v be the radius of the central fringe, v will be equal
to the semi-difference of the quantities ah (h + c)" 1 and ah (h — c)" 1 ,
whence

v =

ach

W r?-

(24).

Having found the centre of the system of circles and the pro¬
jection of the image, or the point where the line joining the eye
and the image cuts the mirror, describe a circle passing through
this projection. This will be the central line of the bright fringe
of the order 0, and its radius will be equal to v. Now describe a
pair of circles whose radii are to v as V(^o ± 1) to *Jn 0 . These will
be the central lines of the two bright fringes of the first order, for
the particular colour to which the assumed value of X relates.
The central lines of the two bright fringes of the second order will
be a pair of circles with radii proportional to \J(n Q + 2), and so on.
The fringes will be broader on the concave than on the convex
side of the central white fringe. When the fringes become
straight, n 0 becomes infinite, and the system becomes symmetrical
with respect to the central fringe. This agrees with observation.

13. When the luminous point is situated in a line drawn
through the eye perpendicular to the mirror a — 0, and we have
simply

7l > _ (° 2 ~ y ) *
- ficV

+ */)•

In this case the achromatic line of the system is reduced to a point,
and the rings are analogous in every respect to the transmitted
system of Newton's rings. For the bright ring of the first order
R=±X, and therefore the radius of the ring is equal to

V /uX . ch

which becomes infinite when c = h. Hence if the luminous point
be at first situated in front of the eye, and be then conceived to
move backwards through the eye till it passes behind it, the rings
will expand indefinitely, and so disappear, and will reappear again
when the luminous point has passed the eye.

ON THE COLOURS OF THICK PLATES.

175

14. All the preceding conclusions agree perfectly with ex¬
periment, so far as qualitative results are concerned; for I have
not taken any measures. The change in the direction of cur¬
vature, which I had not noticed till it was pointed out by the
formula, may be readily seen, when the flame of a candle is the
source of light, by holding the candle at arm’s length nearly in
front of the eye, but a little to the right, then drawing it back
beside the eye, and finally holding it at arm’s length behind the
head, and as nearly in a line drawn through the eye perpendicular
to the mirror as the shadow of the head will allow.

When the candle is held near the eye, a portion only of the
circles can be seen; the circles are in fact reduced to circular arcs,
and these arcs may even become perfectly straight. But when
the candle is placed at a good distance from the eye, suppose
half-way between the eye and the mirror, and a small piece of card
is placed as a screen in front of the flame to keep off the glare of
the direct light, the circles, or at least several of them, may be
seen complete, except that it must be left to the imagination to
fill them up where they are hid by the screen. In this way the
manner in which the rings open out from the centre of the circles
may be observed, though not for the first ring or two, which open
out while the centre is hid by the screen. Instead of a candle
with a screen, it is better to use the image of the sun in the focus
of a small convex or concave mirror.

15. The conclusion deduced from theory which was mentioned
in Art. 13 cannot, of course, be compared with experiment directly.
But the experiment may be successfully performed by substituting
for either the luminous point or the eye a virtual image. Using
for the luminous point the image of the sun in the focus of a small
concave mirror, at the distance of some feet in front of a plane
mirror of which the surface had been prepared with milk and
water, I placed a-piece of plate glass between the mirrors, inclined
at an angle of about 45°. The greater part of the light coming
from the image of the sun was transmitted through the plate of
glass; and on returning from the large mirror a portion of this
light was reflected sideways, so that the rings could be seen by
reflexion in the plate of glass without obstructing the incident
light. The system of rings thus seen was very beautiful, and there
was no direct light glaring in the eye, and yet no screen to hide

176

ON THE COLOURS OF THICK PLATES.

any part of the system. It was easy to know when the image of
the eye in the inclined plate lay in a line drawn through the
luminous point perpendicular to the plane mirror, by observing
when the image of the luminous point seen by reflexion, first at
the plane mirror, and then at the plate of glass, lay in the centre
of the system of rings. Supposing the image to have this position,
on moving the head sideways the opening out of the rings could
be traced from its very commencement. By moving back the
head so as to keep the image of the luminous point in the centre
of the system of rings, it was easy to try the experiment to which
Art. 13 relates, the virtual image of the eye being thus kept in a
line drawn through the luminous point perpendicular to the
mirror, and the eye moving relatively to the luminous point,
which is as good as if the luminous point had moved while the
eye remained fixed. I found, in fact, that on moving back the
head the rings expanded till the bright central patch surrounding
the image filled the whole field of view, and on continuing to move
back the head the rings appeared again. In the position in which
the central patch filled the whole field of view, the least motion of
the eye sideways was sufficient to bring into the field portions of
excessively broad coloured bands.

Between the system of rings seen when the eye was respectively
nearer to and further from the mirror than when in the position
in which the rings became infinite, there was one difference which
may here be mentioned. So long as the image occupied the
centre of the systems, they were similar to each other; but when
the head was moved sideways, the centre of the circles passed in
the first case to the side of the image towards which the head was
moved, and in the second case to the contrary side. This affords
another way of comparing with experiment the result of theory
already mentioned relating to the direction of curvature, and it
will be readily seen that the result of experiment agrees with the
prediction of theory. For, suppose the distance of the eye from
the oblique plate less than that of the luminous point, so that the
virtual image of the eye lies between the luminous point and the
mirror, and let the eye move to the right. Then its virtual image
moves to the left, and therefore, according to theory, the centre
of curvature ought to fall to the left of the image of the luminous
point, right and left being estimated with reference to an eye

ON THE COLOURS OF THICK PLATES.

177

supposed to occupy the position, of the virtual image of the actual
eye. But on account of reflexion at the glass plate, there is
reversion from right to left, and therefore to the eye in its actual
position the centre of curvature falls to the right of the image of
the luminous point, which agrees with observation. The ex¬
periments described in this article may be tried very well with the
flame of a taper, but in examining what becomes of the rings when
they expand it is more satisfactory to use sun-light.

16. In describing the disappearance of the rings, I said that
the central spot expanded till it filled the whole field. In truth,
when the rings had expanded a faint luminous central spot of
finite size remained visible, which was surrounded by a dark ring,
and then a faint luminous ring. It would have been more correct
to speak of the dark ring as faint, since these rings consisted
merely in slight alternations of intensity in a generally bright
field. These rings, however, had evidently nothing to do with the
former rings, which had disappeared; for they continued to have
the image of the luminous point for their centre when the head
was moved to one side. They were doubtless of the same nature
as those which are seen when a luminous point, or the flame of a
candle, is viewed through a piece of glass powdered with lyco¬
podium seed, and arose from the interference of pairs of streams of
light which passed on opposite sides of the globules of dried milk.
I merely mention these rings lest any one in repeating the ex¬
periment should observe them, and mistake them for something
relating to the colours of thick plates.

17. The formula (21) determines the breadths of the several
fringes, which are unequal, except in the case in which the eye
and the luminous point are at the same distance from the mirror.
It will be convenient, however, to investigate a simple formula to
express what may be regarded as a sort of mean breadth. Let the
mean breadth be defined to be that which would be the breadth
of one fringe if the rate of variation of the order of a fringe, for
variation of position in a direction perpendicular to the length of
a fringe, were constant, and equal to the rate in the neighbour¬
hood of the projection of the image, and let /3 be this mean
breadth.

Putting y = 0 in (21), differentiating on the supposition that

12

s. III.

178

ON THE COLOURS OF THICK PLATES.

B and x vary together, and after differentiation putting ah (h + c)' 1
for x, we find

dR = ~dx;
fieri

and since, according to the definition of ft, X l dR = ft 1 dx , we have

/3 =

fichX
2 ta

(25).

When c = h the bands are straight, and of uniform breadth,
that breadth being equal to ft ; and when the bands are not very
much curved ft may still be taken as a convenient measure of the
scale of the system; but the formula (25) is not meant to be
applied to cases in which the projection of the image of the
luminous point falls at all near the centre of the circles.

Section III.

Rings formed by a curved mirror , and viewed directly by the
eye , when the luminous point and its image are not in the same
plane perpendicular to the axis.

18. The rings and bands of which the theory has been con¬
sidered in the two preceding sections may be regarded as forming
the two extreme cases of the general system. In the first case,
the rings appear to have a definite position in space; in the second
case, everything depends upon the position of the eye. These are
the cases of most interest, but there are some properties of the
general system which deserve notice.

In order that rings may be thrown on a screen, it is necessary
that the retardation of one of the interfering streams relatively
to the other should be sensibly constant over the whole of the
dimmed surface, or at least over a large portion of it. But when
the rings are viewed directly by the eye, we are concerned with so
small a portion of the dimmed surface, in viewing a given point of
a ring, that the rings may be seen very well in cases in which
they could not be thrown on a screen. Moreover, we have seen
that even independently of the small size of the pupil, a portion
at least of the system is seen distinctly when the image of the

ON THE COLOURS OF THICK PLATES.

179

luminous point is seen distinctly. Omitting further consideration
of the conditions of distinctness, let us regard the eye as a point,
and investigate the form and character of the rings.

19. Let /, g , h be the co-ordinates of the eye. To find the
retardation, it will be sufficient, as in Art. 9, to write f g , h for
a\ b\ c, and take %, y to denote the co-ordinates of that point of
the mirror on which a given point of a ring is seen projected. The
whole retardation is the sum of the expressions in (5) and (11);
and making the above substitution we find

((m + by)\ ...(26).

- “ {is (s - is) + mr- ~) <«+«} ■ -m-

Hence the bands still form a system of concentric circles. If
Y f Y be the co-ordinates of the centre of the system,

V h \h pj c\o p) 7

'" G-l)(i-?-D.

and F may be written down from symmetry.

The equations of a line joining the eye and the luminous point

£ — 1 “"^— £ ~~ c
f—ct g — b h — c‘

At the point in which this line cuts the mirror £* = 0 , or at least
is a very small quantity, which may be neglected. Hence we
have

5 _/

? = f-T .( 28 ),

from whence rj may be written down if required. If £ 1? rj x denote
the co-ordinates of the point in which the line joining the eye and
the image meets the mirror, £ 1? 97 1 may be obtained from rj by
writing a x> b x , c x for a, 6 , c, where a v b x , c x denote the co-ordinates

of the image. Observing that

12—2

180

ON THE COLOURS OF THICK PLATES.

we find

-A

5-rn.«

T H-

hep

The formulae (27), (28), and (29) shew that X is equal to the
semi-sum of £ and and for the same reason Y is equal to the
semi-sum of rj and Hence the geometrical construction given
in Art. 11 for finding the centre of the system in the case of a
plane mirror applies equally to a curved mirror, even when the
curvatures of the two surfaces are different. Since the retarda¬
tion vanishes for the image itself, it follows that the achromatic
line is a circle having the two points of intersection above men¬
tioned for opposite extremities of a diameter.

20 . It follows from the expressions for X and F, or from the
geometrical construction to which they lead, that if the eye be not
in the line joining the luminous point and its image, whenever it
crosses either of two planes drawn perpendicular to the axis, and
passing, one through the luminous point, and the other through its
image, the centre of curvature of the bands moves off to an infinite
distance, and the bands become straight, and then bend round the
other way.

When the eye coincides with the luminous point, f f g, h
become equal to a, b, c, respectively, and 11 vanishes independently
of x and y. The same takes place when the eye coincides with
the image, since in this case

f _ ci g __ b 1 1_2

h c’ h e’ h c ~~ p ’

Hence, when the eye crosses either of the planes above mentioned,
remaining in the line joining the luminous point and its image,
instead of bands which become straight and then change curvature,
we have rings which disappear by moving off' to infinity, and then
appear again.

I have verified these conclusions by experiment, substituting
when necessary a virtual image of the eye for the eye itself, in the
manner explained in Art. 15. The experiments embraced the
following cases, in the description of which 0 will be used to

ON THE COLOURS OF THICK PLATES.

181

denote the centre of curvature of the mirror, F its principal focus,
L the luminous point, and L 3 its image.

Concave mirror: L beyond 0. Eye (1) beyond L ; (2) passing
L\ (3) between L and L 3 ; (4) passing Z 3 ; (5) between L 3 and
the mirror.—Concave mirror: L between 0 and F. Eye (1)
beyond L 3 ; (2) passing L 3 ; (3) between L 3 and L ; (4) passing L ;
(5) between L and the mirror.—Concave mirror: L between F and
the mirror. Eye (1) beyond L ; (2) passing L ; (3) between L and
the mirror.—-Convex mirror. Eye (1) beyond L ; (2) passing L ;
(3) between L and the mirror.

The mirrors employed wore formed, as usual, with surfaces of
equal curvature. When the observation was made directly, there
was no difficulty in determining at which side of the line joining
the luminous point and its image the eye lay, and consequently in
deciding whether the direction of curvature agreed with theory or
not. When the observation was made by reflexion in a plate of
glass, the eye was placed so that its virtual image fell in the line
LL 3 by moving the head till the image of the luminous point was
seen in the centre of the system of rings. The radii of the two
surfaces of the mirror being the same, or only differing by a small
quantity comparable with the thickness of the glass, the surfaces
may be regarded as forming a pair of concentric spheres; and
therefore, everything being symmetrical with respect to the line
joining L and Z a , when the image of the eye is in this line the
bands necessarily become rings, having the image for their centre.
Hence the theory of the rings or bands, which it is the object of
the experiment to compare with observation, is not involved in
the assumption that the image of the eye was in the line LL 3
when the system of rings appeared arranged symmetrically around
the image of the luminous point. By moving the head a little to
one side, and observing whether the centre of the system of rings
then lay to the right or left of the image, it was easy to compare
theory with observation as to the direction of curvature.

There was no difficulty in telling when the virtual image of
the eye coincided with the image of the luminous point, since in
that case the latter image expanded indefinitely. The phenomena
observed offered no direct test of the coincidence of the virtual
image of the eye with the luminous point, except what arose from
the appearance of the bands themselves. I did not think it worth

182

ON THE COLOURS OF THICK PLATES.

while to take any measures, hut contented myself with observing
that when the eye was in the expected position, or thereabouts,
the rings expanded indefinitely when the image was kept in the
centre of the system, and the bands formed when the image was
allowed to pass to one side changed curvature as the head moved
backwards and forwards.

21. The bands may be considered as completely characterized
by the position and magnitude of the achromatic line, and by the
value of the numerical quantity which has been already defined
as the central order. A simple geometrical construction has
already been given for determining the achromatic circle. Sub¬
stituting X, 7 for os, y in (26), and denoting the resulting value of
B by — we find

t \h \c p

\h cJ \h c p

b( 1_1

c \h p

,...(30).

In the application of this formula n 0 is to be taken positively.

Denoting as before the radius of the achromatic circle by v, we
find from (28), (29), and the formulae thence derived which give

V> Vv

When the bands are nearly straight, instead of the central
order it is more convenient to consider the mean breadth of a
fringe. According to the definition of ft,

v+ ftdiiQ : v :: V(^o+ dn^) : \/n 0 ,

since the radii of the several rings arc as the square roots of their
orders. We have therefore

v

2 n 0

(32).

22 . In the case of a concave mirror, if a small flame be so
placed as to coincide with its image, and be then moved a little
towards the mirror, or from it, it is possible to see a single

ON THE COLOU11S OF THICK PLATES.

183

system of rings with both eyes at once, if the eyes be situated
symmetrically with respect to the flame and its image. The rings
so seen appear to be situated between the flame and its image.
Let E be the right and E' the left eye, and suppose the head
so placed that the line LL 3 bisects EE' at right angles. On
account of the similarity of position of the two eyes, the system of
rings seen with one eye must be exactly like the system seen with
the other, and therefore, in order that a single system may be seen
with both eyes at once, it is necessary and sufficient that the axes
of the eyes be directed to the centres of the respective systems.
It has been shewn already that the projected place on the mirror
of the centre of the system seen with either eye, suppose the right
eye, bisects the line joining the projected places of the flame and
its image. On account of the supposed smallness of the obliquities,
this is the same thing as saying that the centre of the system seen
by the right eye appears in the direction of a line bisecting the
angle LEL r Similarly, the centre of the system seen by the left
eye appears in the direction of a line bisecting the angle LE'L Z .
In order therefore that a single system only may be seen, the
axes of the eyes must be made to converge to the point in which
the bisecting lines intersect LL <]} and therefore the system of rings
will appear to be situated between the flame and its image.

Since the angles LEL Z , LE' 1 are bisected by- the axes of the
eyes when the system of rings is seen single, it follows that the
flame and its inverted image are each seen double, in such a
maimer that the erect flame seen by either eye is superposed on
the inverted flame seen by the other. This agrees with observa¬
tion : in fact, I was led by experiment to the above rule for
determining the apparent position of the rings before I had deduced
it from theory. The observation was made when the flame was in
front of its image, in which case the position of the rings in space
appears more definite than when the image is in front of the flame.

184

ON THE COLOURS OF THICK PLATES.

Section IV.

Straight bands formed by a plane mirror at a considerable
angle of incidence , and vieived by the eye , either directly , or through
a telescope .

23. As the angle of incidence increases the bands become
finer and finer, and after they have become too fine to be dis¬
tinguished by the naked eye they may still be seen through a
telescope, provided the source of light be sufficiently small. When
the source of light was the image of the sun in a lens of short
focus, I saw traces of the bands when the angle of incidence was
about 24° 50', but they were not at all well formed beyond an angle
of about 10° 40', after which they began to bo confounded with rays
which shot in all directions from the image of the luminous point.
With a mirror made of thinner glass they would probably have
been visible at a still larger angle of incidence. The theory of
Section II. sufficiently explains their origin and general character;
but inasmuch as the obliquity was supposed small in investigating
the formulae of that section, it may be desirable to obtain an
expression for their breadth, in which no approximation shall be
made depending on the smallness of the obliquity, in order to
meet the case of any future measures which may be taken at a
large angle of incidence.

24. The notation being the same as in Art. 10, we have
merely to employ the equations (17) and (L8), without making
any approximation depending on the smallness of <fi and <£'. The
thickness t may still be supposed small compared with c and h.
Neglecting t for a first approximation, and then substituting in the
small terms the values of tan </>' and sec <£' got from the first
approximation, we find

-ft, = VFT7+ V3F+7* + 2 1 t ^•

V o H~ a

Interchanging c and h } s and u, and subtracting, we get finally

. (33 >-

ON THE COLOURS OF THICK PLATES.

185

25. For the achromatic line jB = (), and therefore s : c :: it : h.
Hence s is to u in the constant ratio of c to and therefore, by a
well-known geometrical theorem, the achromatic curve is a circle,
having its centre in the line L 0 E 0 produced, and cutting this line
in the two points in which it is divided internally and externally
in the given ratio. The latter of these points may be formed by
producing LE to meet L 0 E 0 produced, and the former by prod ucing
LLq till the produced part is equal to the line itself, and then
joining the extremity of the produced part with E. Hence the
construction given in Art. 11 for determining the bright band of
the order zero continues to hold good whatever be the angle of
incidence.

26. In the neighbourhood of the image the bands are sensibly
straight, being arcs of circles of very large radius. To find the
mean breadth of a band, it will be sufficient to suppose the point
P to lie in the line L 0 E iU to differentiate equation (88) making
R, s, and u vary together, while s + n remains constant, replace

^ by ”, and after differentiation take a and s to refer to the
du J /3

small pencil, regarded as a ray, by which the image is seen. If
i be the angle of incidence, we may put after differentiation
s = c tan i, u = h tan i. We thus find

2t (c -I- h) sin i cos ;j i

On account of the largeness of the angle of incidence, the
breadth of the bands is sensibly uniform, and therefore /3 may be
regarded as the breadth of any one band. It is to be remembered
that /3 denotes the linear breadth of a band as seen projected on
the mirror. If we denote the angular breadth by w, we have on
account of the smallness of vr.

/3 cos i _ Xc V fju 2, — sin a i

h sec i t (c -f h) sin 2 i

186

ON THE COLOURS OF THICK PLATES.

Section Y.

On the nature of the defection of the two interfering streams
from the course of the regularly refracted light.

27. It was suggested to me by a friend, to whom I was
shewing some of the experiments described in this paper, that
in order to see bands or rings of the same nature it would probably
be sufficient to dim the two faces of a plate of glass, and view a
luminous point through it. But having prepared the two faces
of a piece of plate glass with milk and water in the same manner
as for mirrors, taking care to treat the two faces as nearly as
possible alike, on viewing a luminous point through the plate I
found not the slightest trace of the rings or bands, whatever were
the distance of the eye from the plate. There were indeed one or
two indistinct rings surrounding the luminous point; but these
were of a totally different nature, being analogous to the rings
seen with lycopodium seed, and arising from the interference of
pairs of streams which passed on opposite sides of the milk
globules. There was no difficulty in distinguishing them from the
system sought for, since they continued to have the luminous
point for their centre when the plate was inclined to the line
joining that point with the eye. The absence of rings or bands
indicates therefore that the streams scattered at the opposite sides
of a plate are incapable of interfering.

The rays scattered so as to make infinitely small angles with
the regularly refracted rays belong to a point in the bright band
of the order zero, and are therefore brought to a focus on the
retina when the luminous point is seen distinctly. The same
must be at least very approximately true for neighbouring points
of the system of rings, did any such exist, and therefore a portion
at least of the system would be seen distinctly when the luminous
point was seen distinctly. The distances of the luminous point
from the glass plate, and of the glass plate from the eye, were
comparable with the corresponding distances in the experiment
with a plane mirror, and the thickness of glass was comparable
likewise; and with a mirror the bands are seen with the utmost
facility within wide limits of the thickness of the glass, and of the
distances of the luminous point and of the eye from the mirror.

ON THE COLOURS OF THICK PLATES.

187

But to prevent any doubt as to whether the bands might not
have been too small to be seen when formed by transmission, I
have calculated the retardation in the same manner as in Art. 10.
The result is

x—nhafv+a

where R is the retardation of the stream scattered at emergence
relatively to that scattered at entrance, c is the distance of the
luminous point from the plate, h that of the plate from the eye, t
the thickness, and p, the refractive index of the plate, and cc, y are
the co-ordinates of the point in which the plate is cut by any small
pencil (regarded as a ray) which enters the eye, and are measured
from the point in which the plate is cut by a line joining the
luminous point and the eye, a line to which the plane of the plate
is supposed to be perpendicular. On substituting numerical values
in the above formulae, it appeared that the dimensions of the rings
were such that they could not possibly have escaped notice had
they really been formed. The non-appearance of the rings leads to
the following law.

In order that two streams of scattered light may he capable of
interfering, it is necessary that they should be scattered, in passing
and repassing, by the same set of particles. Two streams which
have been scattered by two different sets of particles, although they
may have come originally from , the sanie source, behave with respect
to each other like two streams coming from different sources.

According to this law, in all calculations relating to the colours
of thick plates, we must consider the elementary system of rings
or bands corresponding to each element of the dimmed surface of
the mirror, and then conceive these elementary systems superposed.
We must not compound the vibrations corresponding to streams
which have been scattered by different elements, and then find the
resulting illumination.

28. The reason of this law will bo apparent if it be considered
that particles of dust, &c. small as they may be, are usually large
in comparison with waves of light, so that the light scattered at
entrance, taken as a whole, is most irregular; and the only reason
why regular interference is possible at all is, that each particle of
dust acts twice in a similar manner, once when the general wave is
going, and again when it is returning.

188

ON THE COLOURS OF THICK PLATES.

To examine more particularly the mode of action, let P be any
particle of dust, and consider a wave of light which emanates from
any particular element of the flame or source of light whatever it
be. When this wave reaches P and proceeds along it, a portion is
reflected externally in all directions, and with this we have nothing
more to do. When the wave has just passed P, we may conceive
it as having in a certain sense a hole in its front, corresponding in
size to P, that is to say, there will be a certain portion of the
surface forming the general front of the wave where the ether is
quiescent. As the wave proceeds, the disturbance diverges from
the neighbourhood of this hole by regular diffraction, and when
the disturbance reaches the quicksilvered surface the general wave
suffers reflection, as well as the secondary waves, which, having
diverged from the neighbourhood of P, do not constitute a wave
with an unruffled front, in consequence of the absence of secondary
waves diverging from the hole, which would be necessary to com¬
plete a wave with a front similar to that of the original wave. If
we consider any particular diffracted ray, the chances are that
on its return it will get out by regular refraction, since the dust is
supposed to cover a moderate portion only of the first surface of
the mirror. A portion of the original wave which entered the
glass by regular refraction at a certain distance from P, after
regular reflexion is incident on P from within. The chances are
that the portion thus incident on P does not correspond to a spot
where the front of the wave is materially ruffled by diffraction at
entrance, so that in considering the wave incident on P we may
neglect the previous diffraction. The wave, then, just after
refraction, is incident on P, by which a portion is reflected back
again in various directions, with which we have nothing to do,
a portion, it may be, is refracted or absorbed by P, and the
remainder passes on. The wave so passing on diverges from
the neighbourhood of P by ordinary diffraction, and the two
diffracted streams, having been diffracted in a similar manner by
the same particle, are in a condition to interfere. The similarity
of the two diffractions will be considered in more detail in the
next section.

Now while the light is still in the glass conceive the particles
of dust removed, and then replaced at random. The chances are
that no particle will now occupy the position formerly occupied by

ON THE COLOURS OF THICK PLATES.

189

P. Let P' be the particle nearest to the former position of P;
and, to make a supposition as favourable as possible to interference,
let P' be the very particle P moved a little along the surface
without rotation. Although the interval of retardation R of the
two streams diffracted by P in its first position, and reaching a
given point of space, is sensibly the same as the interval of
retardation of the two streams diffracted by P in its second
position, and reaching the same point, yet this interval would be
changed altogether were the transference of position to take place
during the interval of time which elapses between the departure
of the wave from P and its return after reflexion, as may very
readily be seen. The amount, too, by which the interval of
retardation would be changed would vary in an irregular manner
from one particle to another, and therefore no regular interference
would take place. Now the purely ideal case just considered is
precisely analogous to the case of actual experiment when a
luminous point is viewed through a plate of glass with both faces
dimmed, since the particles on one face have no relation to those
on the other. We ought not therefore in such a case as this to
expect to see rings or bands.

29. According to the formula (35), the angular breadth of one
of the bands formed by a plane mirror becomes considerable when
i becomes nearly equal to 90°, so that, apparently, bands ought to
be visible at a large angle of incidence. But if the courses of the
two streams scattered by the same set of particles be traced, it will
be found that they are so widely separated that, for various
reasons, no regular interference can be expected to take place.
Accordingly, the bands are not seen at a large angle of incidence.

30. In the preceding sections I have spoken of the light by
which the rings are formed as having been Mattered at the dimmed
surface. And so if really is, if by that term we merely under¬
stand deflected from the*, course if would have followed according
to the regular law of refraction. But according to the explanation
given in the preceding article the light is not scattered, in the
strict sense of the term, but regularly diffracted. Scattered light
is, strictly speaking, such as that by which objects arc commonly
seen, or again, such as that which is transmitted through white
paper and similar substances. The preceding view of the nature

190

ON THE COLOURS OF THICK PLATES.

of the light by which the rings are formed is confirmed by the
results of several experiments.

In the experiments of Sir William Herschel and M. Pouillet
mentioned in the introduction, as well as in some of those of the
Duke de Chaulnes, rings of the same nature as those formed by a
tarnished mirror of quicksilvered glass were produced in cases in
which the deflection of the light from its regular course was in¬
contestably of the nature of diffraction. From the similarity of
effect we have a right to infer a probable similarity of cause, unless
such a supposition should entail some peculiar difficulty, which
does not seem to be the case in the present instance, but quite
the contrary.

31. Having cleaned the surface of a concave mirror which had
been employed in forming the bands, I rubbed a little tallow on it,
and then wiped the mirror in one direction with a handkerchief,
so as to have a finely striated tarnish on it. The tarnish was not
sufficient to cause much obscurity; but the image of a candle seen
in the mirror was accompanied, as is usual in such cases, by two
tails of light, which ran out in a direction perpendicular to the
striae. Having placed a small flame in the centre of curvature
of the mirror, I found that the rings were formed with great
brilliancy where they were intersected by the tails of light, but
elsewhere they were almost wholly invisible.

Now the tails of light are known to be a phenomenon of
diffraction: the striated tarnish may in fact be regarded as a sort
of irregular grating, and the tails of light are of the nature of
Fraunhofer’s spectra. If a tarnish in general were capable of
producing rings independently of diffraction, there appears no
reason why a tarnish of tallow should not be capable; for the
particles of tallow are as fine as those of most other kinds of
tarnish. But if in the case of a tarnish of tallow the deflection of
the light from its regular course be not a phenomenon of diffrac¬
tion, there appears no reason why the rings should be confined to
the tails of light in the experiment described above.

32. The phenomena of polarization seem however to lead to a
crucial experiment for deciding whether the deflection of the light
from its regular course, which enables the rings to be formed, be
a phenomenon of diffraction, or of scattering in the strict sense of

ON THE COLOURS OF THICK PLATES.

191

the term. When polarized light is scattered, as for example when
it is reflected from or transmitted through white paper, it loses its
polarization, but when polarized light suffers regular diffraction it
retains its polarization.

Having placed a small flame near the centre of curvature of a
concave mirror, of which the surface had been prepared with milk
and water, I placed a NicoPs prism close to the flame, so as to
polarize the light incident on the mirror. On examining the rings
with another Nicol’s prism, they proved to be perfectly polarized.

33 . It may not be considered out of place here to point out
what appears to be the cause of a phenomenon observed by M.
Pouillet. In an experiment in which rings were occasioned simply
by the straight edge of an opaque body held in front of a metallic
speculum, it was found that they were formed distinctly in only
one half of their circumference. The reason of this appears to be
simply as follows. As the waves of light pass the diffracting edge
in their progress towards the mirror, those rays which are diffracted
inwards, so as to enter the geometrical shadow, after being
regularly reflected at the mirror fall upon the opaque body, by
which they are stopped. As these rays are required for the
formation of that half of the system which lies on the same side
as the opaque body, the other half only is well formed. The first
half may be formed obscurely by a few rays which are diffracted
in the required direction at such a distance from the edge that on
their return they pass clear of the edge, and so proceed to interfere
with other rays diffracted by the edge on the return of the general
wave.

Section VI.

Investigation of the angles of diffraction.

34. Something yet remains to be done in order to complete
the theory of these rings and bands, namely, to compare the
two diffractions which a wave of light experiences at its entrance
into the glass and on its return, respectively. For the phase and
intensity of a ray diffracted in a given direction depend altogether
on the circumstances under which the diffraction takes place; and

192

ON THE COLOURS OF THICK PLATES.

were these circumstances materially different in the case of the
two diffractions above mentioned, the rings might be modified, or
might even disappear altogether.

Let us consider first the case of a concave mirror when the
luminous point and its image are in the same plane perpendicular
to the axis. In this case, if we consider any point P on the
dimmed surface, and any point M in the plane of the rings, the
angle of diffraction for the ray diffracted at emergence will be
i 3 PIf*. For the ray diffracted at entrance, the angle of diffraction
measured in air will be LPM S , that is to say, M Z P is the course of
a ray in air which by regular refraction into glass would be
brought into the direction of the ray diffracted at P. If G be the
intersection of the axis and the plane of the rings, G will be the
centre of the system, and the middle point of both the lines LL Z
and MM Z , and therefore LM Z will be equal and parallel to ML Z .
Hence, on account of the smallness of the obliquities, the angles of
diffraction LPM Z , L S PM are sensibly equal, and their planes
sensibly coincident, but the deviations take place in opposite
directions. But between the two diffractions the light undergoes
reflexion; and since the mutual inclination of two rays is reversed
by reflexion, we must conceive the direction of deviation reversed
in the first diffraction, in order to compare the circumstances of
the two diffractions. Allowing for this reversion, we see that not
only are the angles of diffraction sensibly coincident, but the
directions of deviation are the same.

Accordingly, the interference connected with diffraction, and
the interference which gives rise to the colours of thick plates,
take place independently of each other. For, let /, P denote-
the vibrations at M due to two streams of light diffracted by any
particle of dust P on entering the glass, and passing on opposite
sides of P; let J, J' denote the vibrations due to two streams
diffracted at emergence, and passing on the same sides of P as
I, I\ respectively ; and let / -t- P denote the resultant of I and P,
and similarly in other cases. Let % be the difference of phase
corresponding to the retardation P, and w the difference of phase
of J, P, and therefore also of P, P, on account of the similarity of

* In speaking of angles of diffraction, such as L S PM, I shall distinguish between
LJPM and MPL 8 , using the former notation to denote that the deviation takes
place from PL. A to PM, and the latter to denote that it takes place from PM to PL.

ON THE COLOURS OF THICK PLATES.

193

the two diffractions. We may represent the phases of the four
vibrations by # + % + «, 0 + %, 0 + 0, respectively. Writing

down for greater clearness the phases along with the symbols of
the vibrations, we may express the resultant of the whole four
vibrations by

(I d+x+oi 4* I'd+x) + ^*)•

Moreover, on account of the similarity of the two diffractions, the
coefficients of the two vibrations 7, J may be supposed equal to
each other, and likewise those of the vibrations 7', J\ It is true
that the diffractions take place at different distances from the
source of light, on account of the finite thickness of the glass, but
the difference of distance compared with either of the absolute
distances is a small quantity of the order t , which may be
neglected. Hence the two resultants 7 + 7', J+J' belong to a
diffraction ring of the same kind, and in fact differ in nothing but
in phase; the phase of the former exceeding that of the latter by
Hence the two kinds of interference go on independently of
each other. It is true that in the preceding reasoning we have
considered only two interfering streams 7, 7', and that in calcula¬
tions of diffraction we have to consider the resultant of an infinite
number of streams. But the same reasoning would evidently
hold good whatever were the number of streams 7, 7', 7"... with
their correspondents J , J\ J" ...

35. When an irregular powder, or anything of the kind, is
used to scatter the light, no diffraction rings are visible, because a
given point M in the plane of the rings would belong to a diffrac¬
tion ring of one kind so far as one particle of dust was concerned,
and to a diffraction ring of another kind so far as another such
particle was concerned; and therefore nothing is seen but the
interference rings belonging to thick plates. But when lycopodium
seed is used the lycopodium rings and the interference rings are
seen together. The former are always arranged symmetrically
around the image, as ought to be the case, since they depend only
on the angle of diffraction, which is the same for all points of a
circle described round 7 S as a centre. By this circumstance they
are at once distinguished from the latter, the centre of which falls
half way between the luminous point and its image. On scattering
some lycopodium seed on a concave mirror, and placing a small
flame near the centre of curvature, at such a distance laterally

13

s. III.

194

ON THE COLOURS OF THICK PLATES.

that the two systems of rings intersected each other, I found in
fact that whatever colour appeared in that part of a lycopodium
ring which lay outside the interference system was predominant
in the latter system throughout the remaining part of a circle
described round the image. When the flame was placed in the
axis, an abnormal inequality in the brilliancy of the rings of the
interference system became very apparent. This inequality was
easily seen to correspond to the alternations of intensity in the
lycopodium system.

36. Let us now turn to the general case, in which the luminous
point and the eye are supposed to have any positions, either in the
axis or not far out of it.

The equations of the lines PL :)> PE are

g-ff'a = v-K ^

b. A -y c 3 ’

VjZLff = Z- h

f-x g-y h

Let the small angle L Z PE be projected on the planes of zx and
zy , and let a, ft be the projections, measured positively towards
x, y , and from PL. towards PE. The preceding equations give

x — tu x — f

« =--y ,

c ;! h

which becomes, when a. and c : > arc expressed in terms of a and c,

1

c

1\ a f

d‘ e+ o + i

(3C>

If a, ft' be the projections of the angle of diffraction LPE 3 ,
(where E s denotes the image of the eye,) we may find a', ft' from
a, ft by interchanging a, b, c, and /, g , h, and changing the sign.
If now we change the signs of the resulting expressions, in order
to allow for reflexion at the back, and so compare the circum¬
stances of the two diffractions, we shall obtain the very same
expressions as at first, since (36) and the corresponding expression
which gives ft remain unchanged when a, b , c and f, g } h are
interchanged. Hence in the general case, as well as in the par¬
ticular case first considered, the two diffractions take place under
the same circumstances, and therefore the interference rings are
not affected by any irregularities which may attend the mode

ON THE COLOURS OF THICK PLATES.

195

of diffraction. Furthermore, should the diffraction take place with
a certain degree of regularity, as in the case of lycopodium seed,
so as to exhibit rings or fringes in the aggregate effect of all the
particles which send light into the eye in such a direction as to be
brought to a given point on the retina, the diffraction rings and
the interference rings are seen independent of each other*

37. If 8 be the small angle of diffraction, S 2 = a 2 + /3 2 , whence
from (36) and the other equation which may be written down
from symmetry,

l _ J _ s) a:+ ; + {} + {^ _ r

s> + ; + !F < 87 >-

Hence the loci of the points for which the angle of diffraction
has given values form a system of concentric circles. Referring to
(29), ^e see that the co-ordinates of the centre of the system are
% v t) 1 , so that the centre is situated in the point in which the
mirror is cut by the line joining the eye and the image of the
luminous point. This result might have been foreseen, since 8
vanishes only for the regularly refracted light, and this enters the
eye only in the direction of the line joining the eye and the image.
By introducing the co-ordinates the equation (37) may be

put under the form

S 2 = g + J - 2 ) 2 {(.*-^ )2 + ( 2/-^ )2 }.(38).

Since the diffraction becomes very sparing when the angle of
diffraction becomes at all considerable, it follows that the inter¬
ference rings are but weak at a considerable angular distance from
the imago of the luminous point. This agrees with observation.
In the experiment in which a flame is placed in the centre of
curvature of a concave mirror, and is then moved to one side,
although the rings are symmetrical with respect to the flame and

* From some rough experiments which I have myself made with gauze stretched
in front of a concave glass mirror, of which the surface was clean, I am inclined to
think that the squarish rings observed by the Duke de Chaulnes in the experiment
with muslin, already mentioned, were due to a combination of the coloured rings
of thick plates, and of the appearance produced by a cross-bar grating. If so, the
independence of the two systems would have been rendered evident by slightly
inclining the mirror, when the latter system would have had the image for its
centre, whereas the former would have had for its centre a point situated midway
between the luminous point and the image.

13—2

196

ON THE COLOURS OF THICK PLATES.

its image, so far as regards their forms and colours, they are not
symmetrical so far as regards their intensities, but are decidedly
more brilliant on the side of the image than on the side of the
flame itself. That this is not due merely to the glare of the direct
light, may be proved by holding a small object in front of the
flame, so as to screen the eye from the direct light, when the
rings, though better seen than before in the neighbourhood of the
flame, are still much weaker than on the opposite side, if the
distance of the flame from the axis is at all considerable. For the
same reason, in the case of a plane mirror, when the luminous
point is placed a good distance in front of the eye, so that the
rings do not run out of the field of view, they cannot be traced
throughout the whole extent if the angular distance between the
luminous point and its image be too great, but only throughout
a portion, more or less considerable, on the side of the image.

38. In the case of a concave mirror when the luminous point
is not far from the centre of curvature, and the rings are viewed
by an eye placed at no great distance off, the first factor in the ex¬
pression for 8 2 (equation 38) is not large, and the angle of diffraction
does not increase rapidly in passing away from the image. In the
case of a plane mirror p = oo, and if we suppose c and h equal to
what they were in the former case, or thereabouts, in order to
make the two cases comparable in every respect except the
curvature of the mirror, the factor in question, though larger than
before, is still sufficiently small to prevent S from increasing very
rapidly on receding from the image. Accordingly, in both these
cases, the rings and bands are seen with brilliancy at a considerable
angular distance from the image. But in the case of a convex
mirror of considerable curvature p is negative, and not large, so
that the factor in the expression for S 2 becomes considerable, and
accordingly the angle of diffraction increases rapidly on receding
from the image. I found, in fact, that such a mirror was
peculiarly ill suited for producing rings or bands, inasmuch as
only a comparatively small portion of the system usually seen was
visible, namely, the portion which lay in the immediate neighbour¬
hood of the image.

[From the Report of the British Association for 1851, Part n. p. 14.]

On a new Elliptic Analyser.

After alluding to various methods which had been employed
in investigating experimentally the nature of elliptically-polarized
light, that is to say, the elements of the ellipse described, the
author exhibited and described a new instrument which he had
invented for the purpose. In its construction he had aimed at
being in all important points independent of the instrument-
maker, assuming nothing but the accuracy of the graduation.

The construction is as follows:—A brass rim, or thick annulus,
is fixed on a stand, so as to have its plane vertical. A brass circle,
graduated to degrees, turns round within the annulus, and the
angle through which it is turned is read by verniers engraved on
the face of the annulus. The brass circle is pierced at its centre,
and carries on the side turned towards the incident light a plate
of selenite, of such a thickness as to produce a difference of retard¬
ation in the oppositely polarized pencils amounting to about a
quarter of an undulation for rays of mean refrangibility. On the
side next the eye the brass circle carries a projecting collar,
and round this collar there turns a moveable collar carrying ver¬
niers, and destined to receive a NicoPs prism.

The observation consists in extinguishing the light by a com¬
bination of the two movements. The retarding plate converts the
elliptically-polarized light which has to be examined into plane-
polarized, and this plane-polarized light is extinguished by the
NicoPs prism. There are two distinct positions of the retarding
plate and the NicoPs prism in which this takes place. In each of
these principal positions the retarding plate and the NicoPs prism

198

ON A NEW ELLIPTIC ANALYSER.

may be reversed (i. e. turned through 180°), and the means of the
readings in these four subsidiary positions may be taken for greater
accuracy. The readings of the fixed and moveable verniers in each
of the two principal positions are four quantities given by obser¬
vation, which determine four unknown quantities, namely, (1) the
index error of the fixed verniers, or, which comes to the same, the
azimuth of the major axis of the ellipse described by the particles
of ether, measured from a plane fixed in the graduated circle; (2)
the ratio of the axes of the ellipse; (3) the index error of the
moveable verniers; (4) the retardation due to the retarding plate.
The unknown are determined by the known quantities by certain
simple formulae given by the author.

Let these unknown quantities be denoted by I, tan sr, i, and p,
respectively, the latter being reckoned as an angle, at the rate
of 300° to an undulation. Let li, r be the readings of the fixed
and moveable verniers respectively in one of the principal posi¬
tions, B!, r the corresponding readings in the other; then

r R' + R . r' + r.

1 9 ? ^ 9 >

sin (?•' — r)
sin" (jK' — R ) ;

tan (?■' — r)
tan {li' — R) ’

The author stated that he had made a good many observations
with this instrument for the sake of testing its performance, and
that he had found it very satisfactory. Inasmuch as light is not
homogeneous, the illumination never vanishes, but only passes
through a minimum, and in passing through the minimum the
tint changes rapidly. This change of tint is at first somewhat per¬
plexing ; but after a little practice, the observer is able to point
mainly by intensity, taking notice of the tint as an additional
check against errors of observation. The accuracy of the observa¬
tions is a little increased by the use of certain rather pale coloured
glasses.

To give an idea of the degree of accuracy of which the instru¬
ment is susceptible, suppose the ratio of the axes of the ellipse
described to be about 3 to 1. In this case the author found that
the mean error of single observations amounted to about a quarter
or the fifth part of a degree in the determination of the azimuth,
three or four thousandths in the determination of the ratio of the

ON A NEW ELLIPTIC ANALYSER.

199

minor to the major axis, and about the thousandth part of an
undulation in the determination of the retardation.

On account of the accuracy with which the retardation is de¬
termined, and the largeness of the chromatic variations to which it
is subject, the instrument may be considered as determining, not
only the elements of the ellipse described, but also the refrangibi-
lity of the light employed, or its length of wave, which corresponds
to the refrangibility. The author stated that the error of the thou¬
sandth part of an undulation, to which the determination of the
retardation was subject, corresponded to an error of only the twen¬
tieth or thirtieth part of the interval between the fixed lines
D and E of Fraunhofer.

[Apart from the details of construction of the instrument, the
distinctive feature of the method of observation here described
consists in regarding the retardation due to the plate as an un¬
known quantity, which is eliminated by the method of observation.
The author was not aware when the paper was communicated to
the British Association that he had been anticipated in this by
Mac Cullagh. (See Proceedings of the Roycd Irish Academy , Vol.
II. p. 384, or Collected works of Mac Callagh, p. 239.) Mac Cul-
lagh’s actual observations were made with a Fresnel’s Rhomb, not
a retarding plate, but of course the principle is the same in the
two cases.

In the use of the instrument here described, the observer has
got to extinguish the transmitted light, or in case the incident
light used be not homogeneous, to reduce it to a minimum, by the
alternate employment of the two angular motions of which such an
instrument must be susceptible.

In designing the instrument, the choice of the two angular
motions which shall be independent of each other is not altogether
a matter of indifference. It might have been constructed either
(a) so that the Nicol’s prism and the retarding plate should rotate
independently, or (6) so that the prism and plate should rotate to¬
gether and the prism independently, or (c) so that the prism and
plate should rotate together and the plate independently. It is
evidently desirable that in adjusting one of the rotations so as to
reduce the intensity of the transmitted light to a minimum the
adjustment of the other should not be much put out.

200

ON A NEW ELLIPTIC ANALYSER.

Let 0 be the azimuth of the plane of polarization of the light
transmitted by the Nicol, measured from a fixed direction, (j> that
of one of the neutral axes of the plate when the incident light, sup¬
posed to be homogeneous and perfectly elliptically polarized, is
wholly extinguished, 0 + 80, <j> + 8(j) the actual azimuths in the
course of an observation. In the true positions of the plate and
Nicol, the elliptically polarized light presented for observation
is converted by the plate into plane-polarized, the plane of polari¬
zation having the azimuth 0. In consequence of the errors of
azimuth 80, 8<j>, the light falling on the Nicol has a component
polarized perpendicularly to the azimuth 0 + 80. Let us deter¬
mine the intensity ( Q) of this light as a function of the errors of
pointing.

Let 0, E denote the neutral axes of the retarding plate, N the
plane of polarization of the light transmitted by the Nicol, all
in the true positions, O', E ', N' the same in the actual positions, F
a plane perpendicular to N'. Let us take the intensity of the
elliptically polarized light presented for observation as unity, which
may also be taken as the intensity of the light falling on the
Nicol, whatever be its azimuth, since the small loss by reflection at
the surfaces of the plate is almost rigorously the same for the two
components, and we are only concerned with the relative intensi¬
ties. In the true positions, light of intensity 1 polarized in the
plane N falls on the Nicol; and if we suppose the direction of this
light reversed until it has passed through the plate, substituting
in the plate acceleration for retardation, and then reverse the
direction again, we shall obtain the light presented. This incident
light is now to be supposed to fall on the plate and Nicol in their
actual, not their true positions.

The light, whether reversed or direct, falling on the plate must
be resolved into its components polarized along the neutral axes,
and these again must be resolved so as to retain the components
polarized in the plane F. The four components with which we
are concerned may conveniently be designated as NOO'F, NOEF,
NEOF ', NEE'F , from their successive planes of polarization.
The magnitude of each component will be got by taking the pro¬
duct of the cosines of the successive differences of azimuth; thus
for NOO'F it is

cos ( 0 - </>) cos 8$ cos (0 + 80 + 8$).

ON A NEW ELLIPTIC ANALYSER.

201

The relative retardations of phase for the four are 0, — p, p , 0.
The intensity is found by the usual formula. For the object in
view, 86 and 8cf> may be supposed very small. The result is

Q = ($6 — 8<fi + cos p8(f>y + (sin p cos

where is put for 6 — <fi, the azimuth of the Nicol relatively to
the plate.

When the retardation given by the plate does not differ consi¬
derably from 90°, the first term in Q does not differ greatly from
<ty 2 . The expression for Q shews therefore that it is best that i/r
and (j> } not 6 and <jE>, should be the angles that are altered inde¬
pendently; that is, that of the three constructions mentioned at
the bottom of p. 199, that marked (b) should be the one chosen.
This is the one described at the beginning, and is that with which
the trials of the working of the instrument were made, the
general result of which is mentioned above.

The employment of a plate giving a retardation of about a
quarter of an undulation introduces considerable chromatic varia¬
tions, which we might sometimes desire to avoid. Suppose for
example that we were working with light only slightly differing
from plane-polarized, and did not wish to reduce the intensity by
the use of absorbing media or by selecting a portion of a spectrum ;
it might seem unreasonable to introduce such large chromatic
variations merely to determine a small ellipticity. But in such a
case we are not bound to use a retarding plate such as hitherto
supposed ; we may use a thin plate of mica giving a comparatively
small retardation; all that is requisite being that the retardation
should be large enough to command the ellipticity of the light
that we have to observe, that is, as may readily be shewn, that
it should not be less than 2^, where tan w denotes the ratio of
the minor to the major axis of the ellipse. With the diminished
retardation, the chromatic changes which the retarding plate
introduces into the light observed are of course diminished. The
expression for Q shews that in this case it would be best that the
motions of the plate and prism should be mechanically inde¬
pendent ; but as the instrument does not lend itself to that, we
must take our choice of the two other arrangements, and it may
be shewn from theory that the instrument ought to work well

202

OX A NEW ELLIPTIC ANALYSER.

provided we give to the Xicol, not to the plate, the independent
motion which we have at our disposal.

With a view to its possible employment in such case, the
instrument as originally construrted was furnished with adapting
pieees enabling the Nieol and the plate to exchange places, the
whole instrument being of course in that case turned round, so
that tin* graduation faces the incident light instead of facing the
observer.

If on p. 1!»N / was called the azimuth of the major axis of the
ellipse described, it was merely to facilitate the conception of the
four unknown quant it irs t hat we had to determine. It is hardly
nen>sary to observe that all we are concerned with in the
fonuube is that it is the azimuth off/, principal axis.]

[From the Cambridge and Dublin Mathematical Journal , Yol. vi., p. 215

(. November , 1851).]

On the Conduction of Heat in Crystals.

The 21st, 22nd, and 23rd volumes of the Annales de Ghimie
et de Physique contain three very interesting papers by M. de
Senarmont, describing a series of experimental researches on the
conduction of heat in crystals, as well as in bodies subject to
mechanical pressure in one direction. The mode of experi¬
mental examination employed consisted in cutting a plate from
the crystal to be examined, drilling a small hole through it near
the middle, covering the faces with a thin coating of wax, and
then heating the crystal by a wire or fine tube inserted into the
hole. The heat caused the wax to melt in the neighbourhood
of the hole, and thus a certain isothermal line was rendered
visible to the eye, namely, the line corresponding to the tem¬
perature of melting wax. The variation of conductivity in dif¬
ferent directions was indicated by the elliptical, or at least oval
form of the line bounding the melted wax. This line remained
sufficiently visible after the plate had cooled, and thus the eccen¬
tricity of the ellipse and the azimuth of its major axis could be
examined at leisure. On allowing for errors of observation, it
was found that, for a plate cut in a given direction from a given
crystal, the axes of the ellipse had a determinate ratio, and the
major axis a determinate azimuth. Universally it was found that
the thermic corresponded with the crystallographic symmetry, so
that for example in crystals belonging to the cubical system,
the propagation of heat took place as it would have done in
a homogeneous uncrystallized medium; in crystals belonging to
the rhombohedral system the axis was a direction of thermic

204

ON THE CONDUCTION OF HEAT IN CRYSTALS.

symmetry, and similarly in other cases. When the plate was
not perpendicular to an axis of thermic symmetry, the circum¬
stance was indicated by the non-correspondence of the ovals
formed on the opposite faces, the line joining the centres of the
ovals being in that case oblique to the faces.

The subject of crystalline conduction had previously been in¬
vestigated theoretically by M. Duhamel, in a memoir presented
to the Academy of Sciences in 1828, and printed in the 21st
Galiier of the Journal de CJtJcole Polyteclunque, p. 356. In this
memoir the author deduces the general expressions for the flux
of heat, and the equation of motion of heat, from the hypothesis
of molecular radiation, and applies the general equations to the
solution of a few simple problems, or at least problems which
may very simply ho reduced to the corresponding problems re¬
lating to uncrystallized media. After the publication of the
researches of M. do Senarmont, M. Duhamel was induced to
resume tlus subject, and in a memoir printed in the 32nd
(hthivr of the above-mentioned Journal, he has deduced from
theory a number of general consequences which are directly ap¬
plicable to the experiments of M. do. Senarmont.

In tin* following paper, I propose to present the theory of
crystalline conduction in a, form independent of the hypothesis
of molecular radiation- a hypothesis which for my own part I
regal'd as very quest ionable. The subject will thus be consider¬
ably simplified, for in fact the results flow readily irom certain
very general assumed laws, which no doubt follow as consequences
of the hypothesis of molecular radiation, hut which are of such
simplicity that, the) would seem to follow from almost any rea¬
sonable hypothesis relating to the manner in which the passage
of heat, takes place in thi' interior of a solid body. As regards
the, mat.hemat ieal deduction oi consequences from the general
formula', f have introduced the consideration of what may be
vailed an auxilatnj solid, by which means problems relating to
crystallized bodies are reduced to corresponding problems relating
to ordinary media. All the principal results of M. Duhamel, of
which one at least, was obtained by him in a very artificial
manner, arc 1 , thus rendered almost self-evident, or else directly
reduced to known results relating to ordinary media; and some
results of still greater generality follow with equal facility.

ON THE CONDUCTION OF HEAT IN CRYSTALS.

205

1. Let P be any point of a solid body, homogeneous or hetero¬
geneous, crystallized or uncrystallized; suppose the temperature of
the body to vary from point to point, and let dS be an elementary
plane area drawn through P in a given direction. The quantity of
heat which passes across the element dS in the elementary time
dt will be ultimately proportional to dSdt, and may be expressed
by fdSdt. This quantity / is the flux of heat referred to a
unit of surface. Its value will depend upon the time, upon the
position of the point P, and upon the direction of the elementary
plane drawn through P. For the present, suppose the time and
the position of the point P given, and consider only the variation
of / in different directions about P.

If we suppose the values of f given in the direction of each
of .three planes, rectangular or not, passing through P, its value
in the direction of any fourth plane follows. For make P the
vertex of a triangular pyramid, of which the sides are in the
direction of the first three planes, and the base is parallel to the
fourth, and then conceive the base, remaining parallel to itself,
to approach indefinitely to P. The quantity of heat gained by
the pyramid during the time dt is equal to the quantity which
enters by the faces, diminished by the quantity which escapes by
the base. Now when the pyramid is indefinitely diminished, the
gain of heat in a given indefinitely short time will vary ulti¬
mately as the volume, or as the cubes of homologous lines,
whereas the quantity which passes across any one of the four
faces of the pyramid will vary ultimately as the area of the
face, and therefore as the squares of homologous lines. Hence
in the limit the quantity of heat which escapes by the base will
be equal to the sum of the quantities which enter by the sides,
and consequently if the flux across each be given, the flux across
the base is determinate.

In particular, if we suppose the medium referred to the rect¬
angular axes of x } y, z , and if f x , f yj f z be the fluxes across three
planes drawn through P in directions perpendicular to the axes
of x } y>z \ f the flux across a plane drawn in any other direction
through P; l , m, n the cosines of the angles which the normal to
this plane makes with the axes, we have

f=lf x + mf y + nf z .(1).

206

ON THE CONDUCTION OF HEAT IN CRYSTALS.

This equation shews that if we represent the fluxes across
planes perpendicular to the axes of x, y, z , by three forces or
three velocities, the flux across any other plane will be repre¬
sented by the resolved part of the forces or velocities along the
normal to this plane. Hence the flux across one particular plane
passing through P is a maximum, and the flux across any other
plane is equal to this maximum flux multiplied by the cosine of
the angle between the two planes.

2. Let u be the temperature at P at the end of the time t,
and consider the portion of the solid which is contained in the
elementary volume dxdydz adjacent to P. The quantity of
heat which enters this element during the time dt by the first
of the faces dy dz is ultimately equal to f x dy dz dt, and the
quantity which escapes by the opposite face is ultimately equal
to (f%+ df x /dx.dx)dydz dt. Subtracting the former from the
latter, and treating in the same way each of the other two pairs
of opposite faces, we find that the loss of heat in the element is
ultimately equal to

' d fx + <fyj! + ¥z
Jdx dy dz 4

dx dy dz dt

But if p be the density, p dxdydz will be the mass; and if c be
the specific heat, the loss of heat will also be equal to

— p dxdy dz.ci^dt

ultimately,
we get

Equating the two results, and passing to the limit,

rn = _ (dfx , ( Ve,¥z
” dt \dx ^ dy + dz

( 2 ).

3. The formulse (1) and (2) are general, but for the future
I shall suppose the medium to be homogeneous, and the tem¬
perature to differ by only a small quantity from a certain fixed
standard which we may suppose to be the origin from which u
is measured. Since the medium is homogeneous p is constant*,
and o moreover will be constant, except so far as relates to a
change of specific heat produced by a change of temperature.
But since u is supposed to be small, the terms arising from the

* The expansion of the solid produced by heat is not here taken into account.

ON THE CONDUCTION OF HEAT IN CRYSTALS.

207

variation of c would be small quantities of the second order, since
c only appears multiplied by du/dt, and therefore c may be re¬
garded as constant.

It remains to form the expressions for f x> f y , f z . By the
conduction of heat we mean that sort of communication which
takes place between the contiguous portions of bodies. In the
case of bodies which are partially diathermous, that is to say,
which behave with respect to heat, or at least heat of certain
degrees of refrangibility, in the same way in which semi-opaque
bodies behave with respect to light, or rather in which a green
glass behaves with respect to red rays, heat may be communi¬
cated from one portion of the body to another situated at a
sensible distance. But this is, properly speaking, internal radia¬
tion, and not conduction. Again, if the solid be perfectly diather¬
mous to heat of certain degrees of refrangibility, a portion in the
interior of the mass may by radiation send heat out of the solid
altogether. For my own part I believe conduction to be quite
distinct from internal radiation, although the theory which makes
conduction to be nothing more than molecular radiation and
absorption seems to be received by many philosophers with the
most implicit reliance. No doubt internal radiation may, and I
believe generally if not always does, accompany conduction: and
when the distance which a ray of heat can travel before it is
absorbed is insensible, we may include internal radiation in the
mathematical theory of conduction, and even, if we please, in
our definition of the word conduction. Of course the distance
which we may regard as insensible will depend partly on the
dimensions of the body, partly upon certain lengths relating to
the state of temperature in the interior, and depending upon
the problem with which we have to deal. As an example of a
length of this sort, we may take the distance between consecu¬
tive maxima, if we are considering the internal temperature of a
solid of which the surface has a temperature that is subject to
periodic variations.

A Let us now confine ourselves to conduction, using that
term with the extensions and restrictions above explained. The
temperature u is supposed to be sufficiently small to allow us to
superpose different systems of temperature without mutual dis¬
turbance. If the temperature were the same at all points, there

208

ON THE CONDUCTION OF HEAT IN CRYSTALS.

would be equilibrium of temperature, and the flux at any point
in any direction would be equal to zero. Let then a uniform,
temperature, equal and opposite to that of P, be superposed on
the actual system. Then the temperature at P will be reduced
to zero without any change being made in the fluxes f yy f z .
Hence these quantities will depend, not upon the absolute tem¬
perature at P, but only on its variation in the neighbourhood
of P. Since, by hypothesis, these fluxes have nothing to do with
the temperatures at points situated at sensible distances from P,
they may be assumed to depend only on the differential coef¬
ficients du/dx, dujdy , du/dz , which define the variation of tempera¬
ture in the neighbourhood of P. Since moreover different systems
of temperature may be superposed, it follows that f x , f y , f z are
linear functions of the three differential coefficients above written.
Hence equation (2) may be put under the form

du

c ‘ j s

dx 1 dy

<& 2

where x\ y\ z' } have been written for x , y , z.

5. Let us now refer the solid to the rectangular axes Ox, Oy ,
Oz , instead of Ox\ Oy\ Oz'. Let l , m, n, be the cosines of the
angles x'Ox, x'Oy, x'Oz ; let l' , m', ??', be the same for y\ and
V\ m" n", the same for z'. Then

d -j d d d

dx dx dy dz

But we have also

x = lx -f my -f nz ,

and similar formulae hold good with respect to y' and z\ Since
symbols of differentiation combine with one another according to
the same laws as factors, it follows that the right-hand member
of equation (3) will be transformed exactly as if the symbols of
differentiation were replaced by the corresponding coordinates.
Hence there exists a system of rectangular axes, namely, the
principal axes of the surface,

A'x' 2 + BY + CV 2 + 2 D'y'z' 4 2 E'z'x' 4 2 F'x'y' = 1 .. .(4),

ON THE CONDUCTION OF HEAT IN CRYSTALS.

209

for which the equation of motion of heat takes the form

du . dhi

p d 2 u n d 2 u
' B df + °d?

The system of axes of which the existence has just been
established may conveniently be called the thermic axes of the
crystal. Since the left-hand member of equation (4) is identical
with Ax 2 + By 2 + Cz\ it follows that not only do the principal
axes of the surface (4) determine the directions of the thermic
axes, but the constants A , B , C, are the squared reciprocals of the
principal semi-axes of that surface.

6. Let us now take the thermic axes for axes of coordinates,
and investigate the general expression for the flux of heat. The
general expressions for f xt f y , f z , being linear functions of the
three differential coefficients of u with respect to y> z, will con¬
tain altogether nine arbitrary constants. Substituting the general
expressions in (2), and comparing with (5), we find three relations
between the constants, depending upon the choice of coordinate
axes. These relations being introduced, the expressions may be
put under the following form :

, ^ du du r, du

-f y = B --D iJz + F lTx

, ~du ^ du , -r, du

I shall defer till towards the end of the paper a consideration
of the reasons which make it probable that 1) 13 E l} F x are neces¬
sarily equal to zero. For the present it may be observed that
if the medium be symmetrical with respect to two rectangular
planes, these constants must vanish. For the planes of symmetr}'
must evidently contain the thermic axes; and on account of the
symmetry supposed, if the planes of symmetry be taken for those
of xz and yz , f x must change sign with x, while f y and f z remain
unchanged; and similarly when the sign of y is changed, f y must
change sign, while f x and f z remain unchanged. Referring to (6),
we see that this requires the constants D 13 E v F l to vanish, so that

.00,

s. III.

14

210

ON THE CONDUCTION OF HEAT IN CRYSTALS.

from whence the flux in any direction may be obtained by means
of the formula (1). The formulae (7) contain the expressions for
the flux which result from the theory of M. Duhamel. The con¬
stants A, B, C, denote what may be called the principal con¬
ductivities of the crystal. The reader may suppose for the present
that the following investigations are restricted to media which are
symmetrical with respect to two rectangular planes.

7. It may be worth while to return to the coordinates af 9 y\ z,
which have a general direction, and examine the general expres¬
sions for the flux which correspond to the formulae (7). Putting
f%> fy> fz> for the fluxes across planes perpendicular to the axes
of x\ y f , z\ we get from (1) and (7)

/■ / du du 7V du x

-S‘- A S + F df + p B'

du du j-y, du

r f f'V i 1jV i TV

-fz=0 + L + y

dx

dx f

du

dy'

( 8 ),

where

A' = PA+ m*B + n 2 C\
U = IT A + m'm"B + n'n''C ]

from whence the expressions for B\ E\ and C\ F\ may be written
down by symmetry.

8. So long as we are only concerned with the succession of
temperatures in an infinite solid, we have no occasion to con¬
sider the flux of heat, and the general equation (5) will enable
us to perform all the requisite calculations. In this equation the
indestructibility * of heat is recognized, but not its identity. If
we discard the latter idea, it is nonsense to talk of the heat
gained, we will suppose, by a given element of the solid, as having
come from this quarter rather than from that. If we denote by
A /*, A fy, A f z the quantities by which the values of f Xi f y ,f z given
by (6) exceed those given by (7), we have

d&fx , d ^fy , dbfz _ n
dx dy dz ’

* According to the very important researches of Mr Joule, work is convertible
into heat, from which there can be little doubt that conversely heat is convertible
into work. As regards the present investigation, however, it is perfectly imma¬
terial whether heat be indestructible, or only not destroyed, or rather whether it be
not convertible into anything else, or only not converted.

ON THE CONDUCTION OF HEAT IN CRYSTALS.

211

which is analogous to the equation of continuity of an. incom¬
pressible fluid.

If we suppose all possible systems of values assigned in suc¬
cession to the constants D v E t , F ly the formulas (6) will express
all possible modes of transfer, consistent with our original assump¬
tion respecting the forms of f Xy f y , f z , by which the state of tem¬
perature of the solid at the end of the time t can pass into its
state at the end of the time t 4- dt. Of course, if we suppose heat
to be materia], we cannot help attaching to it the idea of indi¬
viduality. But if we suppose heat to consist in motion of some
sort, which for my own part I regard as by far the more probable
hypothesis, we require a definition of sameness of heat, supposing
we find it convenient to treat the subject in this way. I am not
now going to follow any further the subject which has just been
broached; but I thought it might be worth while to point out
in what manner the additional arbitrary constants found in the
general expressions for the flux beyond what appeared in the
equation of motion, or more properly the equation of successive
distribution , corresponded to an attribute of heat which is neces¬
sarily involved in the idea of a flux, but which is not necessarily
involved in the idea of the successive distributions of a given
quantity of heat.

9. Besides the general equation (5), it is requisite to form
the equation of condition which has to be satisfied at the surface
when the solid radiates into a space at a different temperature.
Let P' be a point in the surface, dS an element of the surface
surrounding P', P'N a normal drawn outwards at P', P a point in
NP f produced, situated at the distance 8 from P'. Consider the
element of the solid bounded by dS, by a plane through P
parallel to the tangent plane at P, and by a cylindrical surface
circumscribing dS , and having its generating lines parallel to
P'N ; and suppose this element to be indefinitely diminished
in such a manner that 8 vanishes compared with the linear di¬
mensions of dS } which is allowable if the curvature at P' be
finite (that is, not infinitely great), as it must necessarily be in
general. The quantity of heat which enters the clement across
the plane through P, as well as the quantity which escapes
across dS, varies ultimately as d&. The area of the cylindrical
surface varies as 8 multiplied by the perimeter of dS, and there-

14—2

212

ON THE CONDUCTION OF HEAT IN CRYSTALS.

fore will vanish compared with dS, since 8 vanishes compared
with the linear dimensions of dS. Hence, even if the quantity
of heat which entered by the cylindrical surface varied as the
surface, it would vanish in the limit compared with the quantity
which escapes by dS. In fact, however, even if we suppose 8
comparable with the linear dimensions of dS , it may be shewn
that the total quantity of heat which enters by the cylindrical
surface is of the order 8dS, because ultimately the quantity of
heat which enters across that portion of the cylindrical surface
for which the flux is positive is equal to the quantity which
escapes across the remainder. Lastly, the gain or loss of heat
by the element during a given time varies ultimately as the
volume of the element. Hence, ultimately, the quantity of heat
which enters the element across the plane through P, during the
time dt, is equal to the quantity which escapes across dS. The
former will be ultimately equal to dSdt multiplied by the value
of the flux obtained from the general formulae (1) and (7) by
taking x , y, z to denote the coordinates of P', and l , m, n, the
direction-cosines of PN. We may assume the latter to be pro¬
portional to the difference u—v between the temperature u of
the solid at the point P and the temperature v of the surrounding
space, and may accordingly express it by h (u — v) dSdt Hence
we have for the required condition,

7 , du r,du ~ du 7 . , , _

IA ^ + mB + nC + h (u — v) = 0.(10).

The quantity h denotes the exterior conductivity of the solid.
It is a certain function of Z, m, n, the form of which M. Duhamel
did not attempt to investigate, nor am I going to attempt the
investigation myself. If however the crystal be covered with a
thin coating of some other substance, sufficient to stop all direct
radiation from the crystal into the surrounding space, h will de¬
pend upon the nature of the coating. In either case h will be
constant throughout any plane face by which the crystal may be
bounded.

10. Let us return to the consideration of the propagation
of heat in the interior of the mass. Imagine the coordinates
x, y, z } of any point altered in the ratios of to V K , to */K,

ON THE CONDUCTION OF HEAT IN CRYSTALS.

213

sJG to fK, where K is constant, and let £, rj, f, be the results.
The equation (5) becomes

This will be true whatever be the value of K , but it will be con¬
venient to suppose that

K* = ABC .(12).

Now imagine a second solid formed from the first by altering
all lines parallel to x in the ratio of V-4 to fK, all lines parallel
to y in the ratio of */B to sJK , and all lines parallel to z in
the ratio of fC to fK, nothing being as yet specified regarding
the nature of the second solid, except that it is homogeneous.
Imagine any number of points, lines, surfaces, or spaces, conceived
as belonging to the first solid, and let the points, lines, &c. de¬
duced from them by altering the coordinates in the ratios above-
mentioned, and conceived as belonging to the second solid, be
said to correspond to the others. On account of the particular
magnitude of K chosen, it is evident that the volumes of cor¬
responding spaces will be equal. Let the second solid be called
the auxiliary solid , and the operation of deducing either solid
from the other, derivation; and suppose the temperatures equal
at corresponding points of the two solids.

The equation (11) shews that the successive distributions of
temperature in the interior of the auxiliary solid will take place
as if this solid were an ordinary medium in which the interior
conductivity bears to K tiie same ratio that the product of the
specific heat and density bears to cp.

The first of equations (7) gives

If now we refer f x , not to a unit of surface, but to that area of a
plane perpendicular to the axis of x which is changed by deri¬
vation into a unit of surfa.ee, we must multiply the above expres¬
sion by \J(BO) and divide it by K. Denoting the result by /$,
using fyj, ft to denote for y, z, what f) denotes for x, and taking
account of (12), we get

, dn du „ du

f‘-- K Tr ) ' = - K <W h= ~ K TK

( 13 ).

214

ON THE CONDUCTION OF HEAT IN CRYSTALS.

It follows from these equations, that if we suppose not only
the temperatures at corresponding points of the two solids to
be always the same, but also equal quantities of heat to flow in
equal times and in corresponding directions across corresponding
surfaces, the flow of heat in the auxiliary solid is what would
naturally belong to an ordinary medium having an interior con¬
ductivity K .

The density of the auxiliary solid being disposable, we may
take it to be the same as that of the given solid, in which case
corresponding spaces will contain equal quantities of matter. It
is only necessary further to suppose the auxiliary solid to be an
ordinary medium having a specific heat c, and an interior con¬
ductivity jST, in order that the motion of heat in the interior of
the two solids should precisely correspond.

11. It remains only to investigate the condition which must
be satisfied relatively to the surface of the auxiliary solid, in
order that the two solids should perfectly correspond in every
respect. Retaining the notation of Art. 9, let dcr be the element
of surface which corresponds to dS ; X, /x, v } the direction-cosines
corresponding to Z, m, n. The quantity of heat which escapes
across dS during the time dt is ultimately equal to h (u — v) dS dt,
and this must be equal to the quantity which escapes across dcr.
Hence it is sufficient to attribute to the auxiliary solid an exterior
conductivity k, such that

But we have

X___ /x __ v __

\JA . I aJB .m \/(J . n \J{A l 2 + Jim* + Cn z )

~ \/(A~ l X 2 -f B~ l [A -f C~ l v z ) ... (15).

Also IdS , XcZcr, are the projections of dS, dcr , on the plane of yz 7
and these projections are proportional to a/(. BO ), K, or to *JK, a /A,
whence we get from (15)

da = j(W+Brrt + W) = x * + B ~ l ^ ^ ■ • -( 1(i )-

The first or second of these expressions will be employed according
as we suppose Z, m > n, or X, v , given.

7 dA .

Ic = t .
acr

.(14).

ON THE CONDUCTION OF HEAT IN CRYSTALS.

215

If the crystal be covered by a thin coating of a given sub¬
stance, h will be constant, and h will be a function of i, m, n, or
of A, fi, v , which is determined by (14) and (16). These formulae
shew that in the case supposed k will have the same value for the
opposite faces of a plate bounded by parallel surfaces.

By means of the auxiliary solid, we may reduce problems
relating to the conduction of heat in crystals to corresponding
problems relating to ordinary media; or, conversely, from a set
of self-evident or known results relating to ordinary media, we
may deduce a set of corresponding results relating to crystals.

12. Let us first regard the crystal as infinite, in which case
the auxiliary solid will be infinite likewise.

In an ordinary medium, if heat be introduced at one point
according to any law, the isothermal surfaces will be a system of
spheres, having the source of heat for their common centre, and
the flow of heat at any point will take place in the direction of
the radius vector drawn from the source. If the temperature be
permanent, and the temperature at an infinite distance vanish,
the temperature at any point will vary inversely as the distance
from the source.

Hence, in a crystal, if heat be introduced at one point ac¬
cording to any law, the isothermal surfaces will be a system of
similar and concentric ellipsoids, having their principal axes in
the direction of the thermic axes drawn through the source, and
proportional to the square roots of the principal conductivities.
The flow of heat at any point will take place in the direction
of the radius vector drawn from the source. If the temperature
be permanent, and vanish at an infinite distance, the temperature
along a given radius vector will vary inversely as the distance
from the source.

It will frequently be convenient to refer to an ellipsoid con¬
structed with its principal axes in the direction of the thermic
axes, and equal to 2y 'A, 2 \JB, 2v^s respectively. I shall call
this ellipsoid the thermic ellipsoid.

13. In an ordinary medium, whether finite or infinite, in
which the temperature varies from point to point, and may be
either constant or variable as regards the time, the flow of heat
at any point takes place in the direction of the normal to the

216

ON THE CONDUCTION OF HEAT IN CRYSTALS.

isothermal surface passing through that point, that is, in a direc¬
tion parallel to the radius vector drawn from the centre of the
thermic sphere to the point of contact of a tangent plane drawn
parallel to the isothermal surface at the point considered.

Now the tangency of two surfaces is evidently unchanged
by derivation. Hence, in a crystal, if we have given the direc¬
tion of the isothermal surface at any point, we may find that of
the flow of heat by the following construction. In a direction
parallel to the isothermal surface at the given point draw a
tangent plane to the thermic ellipsoid, and join the centre with
the point of contact: the flow of heat will take place in a direc¬
tion parallel to this joining line. In other words, the flow of
heat will take place in a direction parallel to the diameter
which is conjugate to a plane parallel to the isothermal surface
at the given point.

14. Conceive a plate bounded by parallel surfaces to be cut
from a crystal, and heat to be applied towards its centre; and
suppose the lateral boundaries sufficiently distant to produce no
sensible influence on the result, so that we may regard the plate
as infinite. In this case the auxiliary solid will likewise be an
infinite plate bounded by parallel surfaces. Now if heat be
supplied according to any law at one point of such a plate,
or at any number of points situated in the same normal, the
isothermal surfaces will be surfaces of revolution, having the
normal drawn through the source or sources of heat for their
axis, and the isothermal curves in which the parallel faces are
cut by the isothermal surfaces will be circles, having their centres
in the points in which the faces are cut by the normal above-
mentioned.

Hence, in a crystalline plate, if heat be supplied according to
any law at one point, or at any number of points situated in a line
parallel to the diameter of the thermic ellipsoid which is conju¬
gate to the planes of the faces, (a line which for brevity I will call
the line of sources ,) any particular isothermal surface will be a
surface generated by an ellipse which has its plane parallel to
the faces, its centre in the line of sources, and its principal axes
parallel and proportional to those of the ellipse in which the
thermic ellipsoid is cut by a plane parallel to the faces. In par-

ON THE CONDUCTION OF HEAT IN CRYSTALS.

217

ticular, the isothermal curves on the two faces are ellipses of the
kind just mentioned*. Hence

(1) If the plate be cut in a direction perpendicular to one of
the thermic axes, the line joining the centres of a pair of ellipses
which correspond on the two faces to a given temperature (such
as that of melting wax) will be normal to the plate. The prin¬
cipal axes of the ellipses will be in the direction of the two re¬
maining thermic axes, and will be proportional to the square roots
of the corresponding conductivities.

(2) If the normal to the plate be not a thermic axis, the line
joining the centres of the ellipses will be inclined to the normal,
its direction being determined as above explained.

(3) If the plate be cut in a direction parallel to either of the
circular sections of the thermic ellipsoid (the three principal con¬
ductivities being supposed unequal,) the isothermal curves on both
faces will be circles, but the line joining the centres of the two
systems of circles will be inclined to the normal.

If the heat be communicated uniformly along the line of
sources, or if there be only a single source situated midway be¬
tween the faces, or more generally if the sources be alike two

* The problem solved in this article forms a good example of the advantage
of considering the auxiliary solid. In M. Duiiamel’s memoir the plate is regarded
as extremely thin, so that the variation of temperature in passing from one point
to another of the same normal may be considered insensible—and it is remarked
that the second case (in which a normal to the plate is not a thermic axis) is
much more difficult than the first; whereas here the plate is not necessarily thin,
and both cases follow immediately from what with regard to an uncrystallized
body is self-evident. M. Duhamel lias shewn that the isothermal curves on the
two faces are ellipses, having their principal axes parallel and proportional to
those of the ellipses in which the thermic ellipsoid is cut by planes parallel to
the faces of the plate: but his demonstration that the line joining the centres
of the two systems of ellipses has the direction assigned in the text does not
seem altogether satisfactory, because the analysis only applies to the case in
which the thickness of the plate is regarded as indefinitely small; whereas the
space by which the ellipse corresponding on one face to a given temperature over¬
laps the ellipse corresponding on the other face to the same temperature is a small
quantity of the order of the thickness of the plate, and ought for consistency’s sake
to be neglected.

The results contained in the remaining part of this paper are not found in the
memoirs of M. Duhamel. It may here be remarked, that the results arrived at by
the consideration of the auxiliary solid, such for example as that of Art. 17, might
have been obtained by referring the crystal to oblique axes parallel to a system of
conjugate diameters of the thermic ellipsoid.

218

ON THE CONDUCTION OF HEAT IN CRYSTALS.

and two, those which belong to the same pair being situated at
equal distances from the two faces respectively, the isothermal
curves belonging to the same temperature in the two systems
will be of equal magnitude, provided that the exterior con¬
ductivity h have the same value for the two faces. The last
condition is satisfied, according to what has been already re¬
marked, when the two faces are covered by a thin coating of
the same substance, which regulates the exterior conductivity;
but it is probable that it may be satisfied generally even if the
faces be left bare, provided that they have the same degree of
polish*.

The experiments of M. de Senarmont bear directly on the
first two cases mentioned above. In the case of crystals which
exhibited three different conductivities, it was found that when
three plates were cut in the directions of the three principal
planes, the ratio of the principal axes of the ellipses formed on
one plate, as determined by observation, agreed very closely with
the result calculated from the ratios which had previously been
determined by observation from the other two plates. An in¬
teresting experiment bearing on the second case is described by
M. de Senarmont in his second paper (p. 187). A rather thick
plate of quartz, inclined to the axis at an angle of 45°, was
drilled in a direction perpendicular to its plane, and heated by
means of a wire inserted into the hole, after its two faces had
been covered with wax. The curves marked out on the two
faces approximated to the two bases of an elliptic cylinder,
symmetrical with respect to the principal plane, and having its
axis inclined towards the axis of the crystal, (which in quartz
is the direction of greatest conductivity,) so as to cross the
wire, which was perpendicular to the plate. The curves how¬
ever were not elliptical but egg-shaped, having their axes of
symmetry situated in the principal plane, the end at which the
curvature was least being that which was nearest to the wire, so
that the blunt ends on the two faces were turned in opposite
directions, the curves being in other respects alike. It will be
seen at once that the symmetry of the curves with respect to
the principal plane, the obliquity of the line joining their centres,

* This result follows readily from the theory of molecular radiation, according
to the suppositions usually made.

ON THE CONDUCTION OF HEAT IN CRYSTALS.

219

and their equality combined with dissymmetry , follow immediately
from theory. We learn too from theory, that in order to procure
ellipses it would be necessary to drill the hole in the direction of
that diameter of the thermic spheroid which is conjugate to the
plane of the plate.

15. Conceive a bar having a section with an arbitrary con¬
tour to be formed from an uncrystallized substance; let heat be
applied in any manner at one or more places, and suppose the
heat to escape again from the surface by radiation. Consider
only those portions of the bar which are situated at a sufficient
distance from the source or sources of heat to render insensible
any irregularity arising from the mode in which the heat is com¬
municated. If the bar be sufficiently slender, we may regard the
temperature throughout a section drawn in a direction perpen¬
dicular to its length as approximately constant, without assuming
thereby that the isothermal surfaces are perpendicular to the
length. Let x be measured in the direction of the length, and
consider the slice of the bar contained between the planes whose
abscissm are x and x 4- dx. Let u be the temperature of the bar
at the distance of the first plane, p the perimeter, and Q the area
of the section, h the exterior conductivity, or rather the mean of
the exterior conductivities in case they should vary from one game-
rating line to another; ltd c, p, K, he the same as before, and put
Q = ap, so that 4a is the side of a square whose area divided by its
perimeter is equal to Qp~ l .

The excess of the quantity of heat which enters during the
time dt by the first of the plane ends of the slice over that which
escapes by the second, is ultimately equal to KQddu/dx *. dxdt.
The quantity which the slice loses by radiation is ultimately equal
to hp u dx dt, if we take the temperature of the surrounding space,
which is supposed to be constant, for the origin of temperatures.
But the gain of heat by the slice is also equal to cpQ dn/dt . dx dt.
Hence we have

cp

da

dt

, r d: ir h
l\ , • a

dx" a

(17).

If we suppose the heat to be continually supplied, and the
temperature to have become stationary, we get from this equation

/ h ,

u=MtT K < r

+ Ne

-j

h

K(d

0«),

220

ON THE CONDUCTION OF HEAT IN CRYSTALS.

where M and JST are arbitrary constants. If now a be small, it
follows from (18) that the lateral flux at the surface, which is
equal to hu , as compared with the longitudinal flux, which is equal
to — K du/dx, is a small quantity of the order f(ha/K).

We may deduce the same consequence for the case of variable
temperatures from the equation (17), without troubling ourselves
with its solution. Conceive any number of bars of different sizes
to be heated in a similar manner, and for greater generality, sup¬
pose the values of c, p, K, and h , as well as a, to be different for
the different bars. Let x } x\ x"... be corresponding lengths, and
t , t\ t" corresponding times, relating to the several bars. The
equation (17) shews that the temperatures at corresponding
points and at the end of corresponding times may be the same
in all the bars, provided

ioc£ f. (19) -

These variations contain the definition of corresponding points
and corresponding times. In order that the temperatures in the
different bars should actually be the same at corresponding points
and at the end of corresponding times, it is sufficient that the
initial circumstances, or more generally the mode of communi¬
cating the heat, should be such as to give equal temperatures
at the points and times defined by the variations (19), which in
this point of view may be regarded as containing the definition
of similarity of heating. Now, in comparing the longitudinal
flux at corresponding points, if we take du the same, dx must
vary as determined by (19), and therefore the flux will vary as
K a~ l h), or f^Ka^h), and the ratio of the lateral flux at

the surface to the longitudinal flux will vary as f(haK ~ 1 ); so
that if we suppose a to decrease indefinitely, h and K being
given, the ratio in question will be a small quantity of the order
filialK) as before, and will ultimately vanish.

The second of the variations (19) shews that if we suppose the
heat to be supplied to one bar in an irregular manner as regards
the time, the fluctuations in the mode of communicating the heat
must become more and more rapid as a decreases, in order that
the similarity of temperatures may be kept up. If the fluctua¬
tions retain their original period, the motion of heat will tend
indefinitely to become what we may regard at any instant as

ON THE CONDUCTION OF HEAT IN CRYSTALS.

221

steady, and thus we fall back on the case first considered. We
may conclude therefore generally, that if the bar be sufficiently
slender, the direction of the maximum flux, even close to the sur¬
face, will sensibly coincide with that of the length of the bar; so
that the isothermal surfaces, which are necessarily perpendicular
to the direction of the flow of heat, will be planes perpendicular
to the axis of the bar.

By supposing the bar to be the auxiliary solid belonging to
a crystalline bar, we arrive at the following theorem. If a slender
crystalline bar be heated at one end, and if we confine our atten¬
tion to points of the bar situated at a sufficient distance from the
source of heat to render insensible any irregularities attending the
mode of communicating the heat, the isothermal surfaces will be
sensibly planes parallel to the diametral plane of the thermic ellip¬
soid which is conjugate to the system of chords drawn parallel to
the length of the bar. These planes will necessarily have an
oblique position unless the direction of the length of the bar be
a thermic axis of the crystal.

The same result might have been obtained without employing
the auxiliary solid, by first shewing that when the bar is suffi¬
ciently slender the direction of the flow of heat sensibly coincides
with that of the length of the bar. We should thus be led to a
problem exactly the converse of that treated in Art. 13, namely,
Given the direction of the flow of heat, to find that of the iso¬
thermal surface.

16. It may be shewn in a similar way, that if a thin plate
be formed of an uncrystallized substance, and be heated at one
or more places, or over a finite portion, if we consider only
those parts of the plate which are situated at a sufficient dis¬
tance from the sources of heat to render insensible any irregu¬
larities attending the mode in which the heat is communicated,
the flow of heat will take place in a direction sensibly parallel
to the plate, and therefore the isothermal surfaces will be cylin¬
drical surfaces whose generating lines are perpendicular to the
plate. It is here supposed that the lateral boundaries of the
plate are situated at a sufficient distance to render their effect
insensible.

Hence, in a thin crystalline plate heated in a similar manner,
the isothermal surfaces, under similar restrictions, will be cylin-

222

ON THE CONDUCTION OF HEAT IN CRYSTALS.

drical surfaces whose generating lines are parallel to the diameter
of the thermic ellipsoid which is conjugate to the plane of the
plate.

17. The state of temperature, under given circumstances, of
a rectangular parallelepiped formed of an uncrystallizcd substance,
may be determined by certain known formulae which it is not
necessary here to describe.

Hence, the state of temperature of a parallelepiped cut from
a crystal in such a manner that its edges are parallel to a system
of conjugate diameters of the thermic ellipsoid may be deter¬
mined by the same formuhe. This parallelepiped will of course
be oblique-angled, except in the particular case in which its edges
are parallel to the thermic axes. It may be remarked that a
parallelepiped for which the state of temperature shall he deter¬
minable by the formula} in question may be cut from a crystal in
a manner quite as general as from an uncrystallized substance.
In both cases the direction of the first edge is arbitrary, and,
when it is fixed on, the plane of the other two edges is deter¬
mined in direction. The direction of the second edge having
been chosen arbitrarily in the plane above mentioned, that of the
third edge is determined.

It does not seem worth while to notice the crystalline figures
derived from spheres, &c., on account of the mechanical difficulty
attending their execution. Besides, the derivation presents no
theoretical difficulty.

Further consideration of the general expressions for the flux.

18. It has been already remarked, that if the crystal possess
two planes of symmetry, the nine arbitrary constants which ap¬
pear in the expressions for the flux in three rectangular direc¬
tions, from which the flux in any other direction may be derived,
reduce themselves to six, and the expressions for the flux take the
form (8). I proceed now to consider what grounds we have for
believing that these expressions, with only six arbitrary constants,
are the most general possible.

In the first place, it may be observed that this result follows

ON THE CONDUCTION OF HEAT IN CRYSTALS.

223

readily from the theory of molecular radiation. In this theory
the extent of molecular radiation is supposed to be very great
compared with the mean interval between the molecules of a
body, so that the body may be treated as continuous. If E , E'
be any two elements of the body, situated sufficiently near one
another to render their mutual influence sensible, it is supposed
that, during the time dt, a quantity of heat proportional to Edt
radiates in all directions from E, whereof E' absorbs a portion
proportional to E'. On the other hand, E' emits and E absorbs
a quantity also proportional, so far as regards only the magni¬
tudes of j E, E, and dt y to EE'dt. The exchange of heat between
E and E' may therefore be expressed by qEE'dt. The quantity
q is supposed to be proportional, so far as regards its dependence
on the temperatures, to the small difference between the tem¬
peratures of E and E'. It will also depend upon the nature of
the body, upon the distance EE, and in the case of a crystal¬
line body upon the direction of the line EE'; but we need not
now consider its dependence upon these quantities. If the length
EE' = s, and if we suppose the extent of internal radiation to be
very small, we may express the difference between the tempera¬
tures of E and E by dajds . s. It follows then from the theory
we are considering, that the total flux of heat arises from the
exchange of heat between all possible pairs of elements, such as
E, E'; the exchange between any pair E, E being proportional
to the rate of variation of temperature in the direction EE', and
accordingly independent of the variation of temperature in other
directions.

Now suppose the body referred to rectangular axes, and let P
be the mathematical point whose coordinates are x, y, z. Con¬
ceive the body divided into an infinite number of infinitely small
equal elements. Let E be the element which contains P, E any
element in the neighbourhood of P, and consider the partial flux
in the neighbourhood of P which arises from the exchange of
heat between all pairs of elements which have the same relative
position as E and E. Through P draw an elementary plane 8,
which it will be convenient to consider as infinitely large com¬
pared with the dimensions of the elements such as E, and con¬
ceive 8 to assume in succession all possible directions by turning
round P. The partial flux across 8 will vary as the number of

224

ON THE CONDUCTION OP HEAT IN CRYSTALS.

points in which the lines, all equal and parallel to EE, which
connect the pairs of elements, cut the plane or as the cosine of
the angle between the normal to 8 and the direction EE\ Let
EE' = s ; let l', m, n, be the cosines of the angles which this line
makes with the axes of x, y, z , and suppose S to be perpendicular
to each of these axes in succession. We shall thus have for the
partial fluxes f x ,fy>fz> quantities proportional to V dujds, m'dujds,
n' dujds, or to

in du , 7 , ,du , v ,du

l -7-+ hn t + In t >
dx dy dz

7 , ,du du , .du

l vi -bni —\- win
dx dy dz

7 , ,du , t du t , 2
l'n' -y- + inn! -r- + n n
dx dy

du
dz *

Hence, the coefficient of dit/dy in the expression for the partial
flux f x is equal to the coefficient of dujdx in the expression for
the partial flux f y , and the same applies to y , and to x. This
being true for each partial flux, will be true likewise for the total
flux, and therefore the general expressions for the flux in three
rectangular directions, with nine arbitrary constants, will be re¬
duced to the form (8), or the general expressions (6), referred to
the thermic axes, to the form (7).

19. Let us further examine some of the consequences which
would follow from the supposition that the expressions for the
flux referred to the thermic axes have the general form (G).
Conceive a crystalline mass, regarded as infinite, to be heated
at one point according to any law, and let the source of heat
be taken for origin. We have seen already that the succession
of temperatures takes place in an infinite solid in exactly the
same manner whether the expressions for the flux have the
general form (6), or the more restricted form (7), and conse¬
quently, in the case supposed above, the temperature at a given
time is some function of

A’W + B-y + C-'z 2 .

If x, y, z, be the coordinates of any point in a line of motion, or
line traced at a given instant from point to point in the direction
of the flow of heat, dx, dy, dz, will be proportional to f x , f y , f z ,
which are given by (6), and in the present case dujdx, dujdy, dujdz,

ON THE CONDUCTION OF HEAT IN CRYSTALS. 225

are proportional to A~'x, B~\j, Q~'z. Hence the differential equa¬
tions of a line of motion are

dx _ dy

x - FJBT'y + Efi-'z ~y- Dfi-'z + F^A~'x

_ & z /nn\

~z-E t A~ l x + Dfi-'y •'

Taking £, y, f, to denote the same quantities as in Art. 10, and
putting for shortness

=®', EJCAy^a", F t (AB)-* = 21 ), .

we get

d % - d V / 22 s

f - «'"»> + ®"£ t? - a'f + ®"'f ~ + a>'y •'‘

Conceive an elastic solid to be fixed at the origin, and to
expand alike in all directions and at all points with a velocity
of expansion unity, so that a particle which at the end of the
time t is situated at a distance r from the origin, at the end
of the time t + dt is situated at a distance rfc+dt). Conceive
this solid at the same time to turn, with an angular velocity ©
equal to aJ( a>' 2 + a>" 2 + a'" 2 ), about an axis whose direction-cosines
are a/coT 1 , ©"gT 1 , ©'"©“h The direction of motion of any particle
will represent the direction of the flow of heat in what we may
still call the auxiliary solid , from whence the direction of the
flow of heat in the given solid will be obtained by merely con¬
ceiving the whole figure differently magnified or diminished in
three rectangular directions.

This rotatory sort of motion of heat, produced by the mere
diffusion from the source outwards, certainly seems very strange,
and leads us to think, independently of the theory of molecular
radiation, that the expressions for the flux with six arbitrary con¬
stants only, namely the expressions (8), or the equivalent expres¬
sions (7), are the most general possible.

20. Let the auxiliary solid be referred to the rectangular
axes of ?/, £", of which the last coincides with the axis to which
co refers. It may be seen immediately, without analytical trans¬
formation, that the differential equations to the lines of motion
will be

_ dr)' _d£

?-co V '- V ' + co? f. ^

S. TIT.

15

226

ON THE CONDUCTION OF HEAT IN CRYSTALS.

Taking polar coordinates r, 8 in the plane of t{, we have

(sin 8 + © cos 8) (cos 8 dr- sin 6 rdd)

— (cos 8 — cd sin 8) (sin 6 dr + cos 6 rdd );

whence a>dr = rdd . (24),

the differential equation of a system of equiangular spirals in
which the angle between the tangent and radius vector is equal
to tan" 1 ®. We have also from (23)

d? (£' — cot]') dg + (vf + <*>%) dy'

T~ (r-^') 2 +(v+®r) 2

f m'+y'dy' tty-v'd?

= (! + «)

whence (1 4- co 2 ) log f' 4- const = log r 4- cod

= log r + co 2 log r from (24). We have therefore

f' ==mr .(25),

where m is an arbitrary constant.

Hence, in the plane of rf, conceive an equiangular spiral
described about the origin as pole, such that the angle between
the tangent and the radius vector is equal to tan _1 ct). Let it
assume all possible positions by turning once round the pole,
and in each position let it be made the base of a cylinder whose
generating lines are parallel to the axis of Conceive also an
infinite number of cones of revolution described with the origin
for vertex and the axis of £' for axis. The system of curves of
double curvature formed by the intersection of the cones with
the cylinders will be the lines of motion in the auxiliary solid,
on the supposition that the constant co does not vanish. To
obtain the lines of motion in the original solid, it will be suf¬
ficient to conceive the whole figure differently magnified or di¬
minished in three rectangular directions, and we shall thus obtain
a clearer idea of the form of the curves, which is the whole object
of the investigation, than would have been derived from the rather
complicated equations got from the integration of equations (20)
in their original shape.

21. It may be observed, in conclusion, that even if there
were reason to suppose that the constants D v E x , F v were not
necessarily equal to zero, it is only among crystals which possess

ON THE CONDUCTION OF HEAT IN CRYSTALS.

227

a peculiar sort of asymmetry, that we should expect to find
traces of their existence. We have seen already that if the
crystal possess two planes of symmetry, these constants can
have no existence. But the crystalline form (taken as an index
of the degree of symmetry of the internal structure) may indi¬
cate the non-existence of these constants even in cases in which
the crystal does not possess a single plane of symmetry. Take
for example quartz, which was one of the crystals employed by
M. de Senarmont in his experiments. In this crystal, not only
is there no plane of symmetry, but a peculiar kind of asymmetry
is indicated by the occurrence of hemihedral faces, as well as by
the optical properties of the crystal. Let three adjacent edges
of the primitive rhombohedron meeting in one of the solid
angles which is formed by three equal plane angles be denoted
by <2, H , I, and let the opposite edges be denoted by Q\ E !, T .
If we suppose the interior structure to correspond to the crys¬
talline form, whatever we can say of the structure with reference
to the edges (?, H, I , we can say with reference to the edges
JET, /, 0, or /, 6r, jET. This shews that the thermic ellipsoid must
be an ellipsoid of revolution about the axis of the crystal, and
that the line to which co refers must coincide with the axis. But
furthermore, whatever we can say of the structure with reference
to Gr, H , J, we can say with reference to G\ T, H\ or T, H\ Gr' } or
H\ G\ l 7 . This requires that co = — co, and therefore co = 0, and
therefore D t = 0, E x = 0, F x = 0.

[From the Transactions of the Royal Society of Edinburgh, Yol. xx. p. 317.]

On the Total Intensity of Interfering Light.

(Extracted from a letter addressed to Professor Kelland.)

[Communicated January 5, 1852. (Proceedings R. S. E.,
Yol. m. p. 98.)]

Pembroke College, Cambridge.

My dear Sir,

* * * *

In reading your paper in the Transactions of the Royal
Society of Edinburgh , vol. xv., p. 315, some years ago, it occurred
to me to try whether it would not be possible to give a general
demonstration of the theorem, applying to apertures of all forms.
I arrived at a proof, which I wrote out, but have never published.
As I think it will interest you I will communicate it. You may
make any use you please of it.

Case I. Aperture in front of a lens; light thrown on a screen
at the focus, or received through an eye-piece, through which the
luminous point is seen in focus.

The expression for the intensity is given in Airy’s Tract,
Prop. 20. If the intensity of the incident light at the distance of
the aperture be taken for unity, and D be the quantity by which
any element of the area of the aperture must be divided in
forming the expression for the vibration, that expression becomes

I)// Sin X + dxdy,

ON THE TOTAL INTENSITY OF INTERFERING LIGHT. 229

the integration being extended over the whole aperture. If it
should be necessary to suppose a change of phase to take place in
the act of diffraction, such change may be included in the constant
B. If, then, I be the intensity,

and if I be the total illumination,

i* 00 /»00

/= Idpdq.

J — 00 J —00

Now,

j///(>> y)dxdyl =JJJJ /O, y)v') dx dy dx' dy',

the limits of x\ y' being the same as those of x , y. Hence,

D 2 / = //// cos ® + ty' — y) dx dy dx dy'.

In the present shape of the integral, we must reserve the
integration with respect to p and q till the end; but if we
introduce the factor where the sign — or -f is supposed to

be taken according as p or q is positive or negative, we shall
evidently arrive at the same result as before, provided we suppose
in the end a and /3 to vanish. When this factor is introduced, we
may, if we please, integrate with respect to p and q first. We
thus get

D 2 J-limit of

-

e Tap*Pq cog

• x + qy' — y) dx dy dx' dy' dp dq.

Now,

| e* a v cos (kp — Q) dp = cos Q I cos kp dp

J -00 j —00

1*00

-f sin Q e* a P sin kp dp
J —00

= 2 cos

€ -ap

cos kp dp =

2a cos Q
a 2 -f/r 2

230 ON THE TOTAL INTENSITY OF INTERFERING LIGHT.

A similar formula holds good for q, whence
D-I = limit of

ffff _4«/3_

dx dy dx' dy'.

Let now

whence

2tt (x! — oo)

bx

= au,,

7 , bXa 7
dx = dUy
Ztt

and the limits of u are ultimately — go and 4- oo , since a ultimately
vanishes. Hence

limit of

2a daf _

'27r (x — x )\ 2

bX

bX r du _
7T J ~oo 1 + U 2

A similar formula holds good for y, and we have, therefore,
L 2 I = b 2 X 2 jjdxdy = b\ 2 A,

if A be the whole area of the aperture or apertures.

Now / ought to be equal to A, and, therefore,

D = bX.

Case II. Aperture in front of a screen.

The formula for the illumination is given in Airy’s Tract,
Art. 73. We have as before,

D 2 / = limit of
ap

Xab \ a ■+• b

aq

X ~aU) + [ y '~f+b) \^ d ddx'd 1 /dpd < 1

= limit of

//////*“”" °“ ^ -*+!/•- rt

■ (a! - x) - ^ <y - y)j dx dy dx dy’ dp dq

ON THE TOTAL INTENSITY OF INTERFERING LIGHT. 231

= limit of

2a

2ir ( x' — x)\ r

X& )

C0S ^ \a~T~ ^ ~ * 2 + ^ ~ dx dy doc '

Now, when a vanishes, the whole of the integral
f 2adx'

<r+ ( —X6“—)

is ultimately comprised between limits for which x r is infinitely
close to x, and similarly with respect to y ; so that ultimately

cos ~ h ~ ^ y "' ~ y ^ =1

within the limits for which the quantity under the integral sign
does not vanish. Hence, passing to the limit, we get

D 2 ! = X 2 b 2 M dxdy = \ 2 b 2 A,

as before.

Case III. Everything the same as in Case II., except that
the phase of vibration is retarded by p, where p is some function
of x and y.

This case is very general. It includes, as particular cases,
those numbered I. and II. The experiment with Fresnel’s mirrors
or a flat prism is also included as a particular case *.

From what precedes, it is plain that we should have in this

D 2 I = limit of

(a + b) _ ,

20

2?r ( y'-y) V'
“ \b J

[. x 2 — x 2 + y 2 -y 2 ]~-p' + p> dx dy dx dy ',

* Thus, in the case of the flat prism, if P, Q be the virtual images corresponding
to the halves AB, BG, if we produce AB

to I), we may suppose the light which ^

falls on BG, instead of coming from Q, to -

come from P, and to have been accelerated ' __ ' [ q

by the passage through the wedge DBG
of air instead of the same wedge of glass.

232 ON THE TOTAL INTENSITY OF INTERFERING LIGHT.

where p is the same function of of and yf that p is of x and y.
The same reasoning as before leads to the same result.

I do not regard the preceding demonstration of a result which
you were the first to announce, as of any physical interest after
what you have yourself done. Still it may not seem wholly
uninteresting, in an analytical point of view, to demonstrate the
proposition for any form of aperture.

Of course, by comparing the result X 2 b' 2 A with that obtained, in
particular cases, by integrating in the straightforward way, we
may arrive at the values of various definite integrals.

I am, dear Sir,

Yours very truly,

G. G. Stokes.

[From the Transactions of the Cambridge Philosophical Society , Vol. ix., p. 399
(read Feb. 16 and March 15, 1852).]

Oh the Composition and Resolution of Streams of
Polarized Light from different Sources.

When a stream of polarized light is decomposed into two
streams which, after having been modified in a slightly different
manner, are reunited, the mixture is found to have acquired
properties which are quite distinct from those of the original
stream, and give rise to a number of curious and apparently
complicated phenomena. These phenomena have now, however,
through the labours of Young and Fresnel, been completely
reduced to law, and embraced in a theory, the wonderful sim¬
plicity of which is such as to bear with it the stamp of truth.
But when two polarized streams from different sources mix to¬
gether, the mixture possesses properties intermediate between
those of the original streams, and none of the curious pheno¬
mena depending upon the interference of polarized light are
manifested. The properties of such mixtures form but an un¬
inviting subject of investigation; and accordingly, though to a
certain extent they are obvious, and must have forced themselves
upon the attention of all who have paid any special attention to
the physical theory of light, they do not seem hitherto to have
been studied in detail.

Were the only object of such a study to enable us to calcu¬
late with greater facility the results obtained by means of cer¬
tain complicated combinations, the subject might deservedly be
deemed of small importance. For the object of the philosopher
is not to complicate, but to simplify and analyze, so as to
reduce phenomena to laws, which in their turn may be made
the stepping-stones for ascending to a general theory which shall

234 ON THE COMPOSITION AND RESOLUTION OF STREAMS OF

embrace them all; and when such a theory has been arrived at,
and thoroughly verified, the task of deducing from it the results
which ought to be observed under a combination of circumstances
which has nothing to recommend it for consideration but its com¬
plexity, may well be abandoned for new and more fertile fields
of research. But in the present case certain difficulties seem
to have arisen respecting the connexion between common and
elliptically polarized light which it needed only a more detailed
study of the laws of combination of polarized light to overcome;
and accordingly the subject may be deemed not wholly devoid of
importance.

The early part of the following paper is devoted to a demon¬
stration of various properties of elliptically polarized light, and
of oppositely polarized streams. When two streams of light are
called oppositely polarized, it is meant that, so far as relates to
its state of polarization, one stream is what the other becomes
when it is turned in azimuth through 90°, and has its nature
reversed as regards right-handed and left-handed. Most, if not
all, of these properties have doubtless already occurred to per¬
sons studying the subject, but I am not aware of any formal
demonstrations of them which have been published; and indeed
some artifices were required in order to avoid being encumbered
in the demonstrations with long analytical expressions. The com¬
bination of several independent polarized streams is next con¬
sidered, and with respect to this subject a proposition is proved
which may be regarded as the capital theorem of the paper. It
is as follows.

When any number of independent polarized streams, of given
refrangibility, are mixed together, the nature of the mixture is
completely determined by the values of four constants, which are
certain functions of the intensities of the streams, and of the
azimuths and eccentricities of the ellipses by which they are
respectively characterized; so that any two groups of polarized
streams which furnish the same values for each of these four
constants are optically equivalent.

It is a simple consequence of this theorem, that any group
of polarized streams is equivalent to a stream of common light
combined with a stream of elliptically polarized light from a
different source. The intensities of these two streams, as well as

POLARIZED LIGHT FROM DIFFERENT SOURCES.

235

the azimuth and eccentricity of the ellipse which characterizes
the latter, are determined by certain formulas, which will he found
in their place.

The general principles established in this paper bear on two
questions of physical interest. Strong reasons are adduced in
favour of the universality of the law, that the two polarized
pencils which a doubly refracting medium of any nature is
capable of propagating independently in a given direction are
polarized oppositely. In strictness, we ought to speak of two
series of waves rather than two pencils; for it is the fronts of
the waves, not the rays, which are supposed to have a common
direction. The other point alluded to relates to the distinction
between common, and elliptically polarized light. It is shewn
that the changes which are continually taking place in the mode of
vibration may be of any nature, and that there is no occasion, in
the case of common light, to suppose the transition from a series
of vibrations of one kind to a series of another kind to be abrupt.

At the end of the paper the general formulae are applied to
the case of some actual experiments, but these applications are
not of sufficient importance to deserve separate mention.

1. Consider a stream of light polarized in the most general
way, that is, elliptically polarized, and propagated through the
free ether. Let the medium be referred to the rectangular axes
of x, y , z, the axis of £ being measured in the direction of pro¬
pagation. Let a and a + 90° be the azimuths of the principal
planes, that is, the planes of maximum and minimum polariza¬
tion, azimuths being measured about the axis of £ from x to¬
wards y. Let the rectangular components of the displacements
of the ether be represented by lines drawn in the planes of
polarization of the plane-polarized streams which these compo¬
nents, taken separately, would constitute. I make this assump¬
tion to avoid entering into the question whether the vibrations
of plane-polarized light are parallel or perpendicular to the plane
of polarization. If we adopt the former theory, the actual lines
in the figures which we are to suppose drawn will represent in
magnitude and direction the ethereal displacements; if we adopt
the latter, the same will still be the case if we first suppose all
our figures turned round the axis of z, in a given direction,
through 90°.

236 ON THE COMPOSITION AND RESOLUTION OF STREAMS OF

Let the co-ordinates x\ yf be measured in the principal planes
whose azimuths are a, a 4- 90°; let j3 be the angle whose tangent
is equal to the ratio of the axes of the ellipse described, the
numerical value of /3 being supposed not to lie beyond the limits
0 and 90°; let v be the velocity of propagation, t the time, X the
length of a wave, and put for shortness,

9

= <p .( 1 ).

Then about the time t , and at no very great distance from a
given point, suppose the origin, we may represent the displace¬
ments belonging to the given stream of elliptically polarized
light by

x' = c cos /3 sin ( cf> + e), y' = c sin /3 cos (cj> + e).(2).

If the light be convergent or divergent, c will depend upon
<£, but for our present object any variation of c arising from this
cause will not enter into account. The value of a , which deter¬
mines the direction in which x is measured, as well as that of
y3, is given by the nature of the polarization. The polarization
is right-handed or left-handed according to the sign of /3. As
to c and e, the phenomena of optics oblige us to suppose that
they are constant, or sensibly constant, for a great number of
consecutive undulations, but that they change in an irregular
manner a great number of times in the course of one second.
The known rapidity of the luminous vibrations allows abundant
scope for such a supposition, since c and e may be constant for
millions of consecutive undulations, and yet change millions of
times in a second. This series of changes, rapid with respect to
the duration of impressions on the retina, but slow compared
with the periodic changes in the motion of the ethereal particles,
is exactly what we might have expected beforehand from a con¬
sideration of the circumstances under which light is produced, so
far at least as its sources are accessible to us; and thus in this
point, as in so many others, the theory of undulations commends
itself for its simplicity.

If c were constant c 2 would be a measure of the intensity,
so long as we were only comparing different streams having
the same refrangibility. But since c is liable to the changes
just mentioned, if we wish to express ourselves exactly, avoiding

POLARIZED LIGHT FROM DIFFERENT SOURCES.

237

conventional abbreviations, we must say that the intensity is
measured, not by c 2 , but by the mean value of c 2 , which may
conveniently be represented by ttt(c 2 ).

2. Let us examine now whether it be always possible to
resolve the given disturbance into two which, taken separately,
would correspond to two elliptically polarized streams of given
nature. For the sake of clear ideas, it may be supposed that
the azimuths and eccentricities of the ellipses belonging to these
two streams are given and invariable, while the azimuth and
eccentricity of the ellipse belonging to the first stream are given
for that stream, but vary from one to another of a set of streams
which we wish to consider in succession.

Let x 1} c l3 &c. be for the first, and x 2> c 2 , &c. be for the
second stream of the pair, what x\ c, &c. were for the original
stream; and resolve all the displacements along the principal
axes of the latter stream. Then, in order that the original
disturbance may be equivalent to the pair, we must have, inde¬
pendently of 4 >,

x x cos (oq — oi) — y l sin (oq — a) + x 2 cos (a 2 — a) \

- y 2 sin (a 2 -«) = «';.^

x 1 sin (oq — a) + y l cos (oq — ot) + x 2 sin (oq — a) f

+ y 2 cos(a 2 -a) = y'. ,

Conceive aq, jq, x 2 , y 2 , x\ and y' expressed in terms of 4> by the
formulae (2) and the similar formulae whereby x 13 jq, &c. are
expressed, and then let the sines and cosines of <£ + e, 4> + € i>
and <^> + e 2 be developed. In order that equations (3) may be
satisfied independently of 0, the coefficients of sin </> and cos </>
must separately be equal to zero, so that each of these equations
will split into two. We shall thus have four equations to deter¬
mine the four unknown quantities c 19 c 2 , e,, and e 2 . For the sake
of shortness, let

c cos e = g, c sin e — h 3

whether the letters be or be not affected with suffixes; and
further put

a 2 - a== W

>•••(4).

238 ON THE COMPOSITION AND RESOLUTION OF STREAMS OF
then our four equations become

cos & cos 7l . y, + sin f3 1 sin 7l . h t + cos /3 2 cos % . g t

+ sin/3 2 sin 7 ,.A 2 = cos /3. y,
cos ft cos 7l . A 2 - sin ft sin 7l . + cos ft cos <y 2 . A 2

- sin /9 2 sin 7s . y 2 = cos £. A,
cos ft sm 7l . y, - sin ft cos 7l . A, + cos ft sin 7a . g 2

— sin ft cos 7a . A 2 = — sin ft. A,
cos ft sm 7l . Aj + sin /ft cos 7l . y, + cos ft 2 sin <y 2 . A s

+ sin /ft cos %. y 2 = sin ft. g.

Multiplying the first and second of these equations by 1
V ( -l) and adding, then multiplying the third and fourth by
~ V (— 1), 1, and adding, and putting generally

g + J~^lh=G . (5)

we have v h

(cos /Sj cos 7 i — J — 1 sin (3 X sin %) G l -f (cos j3 2 cos y 2

_ -y-isin^ sin 7s! ) ft = CO s ft. ft

(sm /ft cos 7l - J - l cos /ft sin 7l ) ft + (sin /ft cos %

— y — 1 cos As s in %) ft = sin ft. (7; j
which two equations are equivalent to the four (4).

1 tX ntt ? g f ° r * hort ° ess Pi’ P*> ?i> % ^ the coefficients in the
left-hand members of equations (6), we have

_ ft_ft q

( h c° s sin ft “ft, shift- ryftosft = ^ • ■ ■ O)-

On substituting for A> q 2 , & c . their values, we find
Pl2i “• Ml = C0S (?! ~ 7 2 ) S ^ n (i8 2 — f3 x )

+ y — 1 sin ( 7i — 7a ) cos (ft, + /ft).
Now the equations (6) cannot be incompatible or identical unless
the above quantity vanish. But this can only take place when

sin (/ft-/ft) = 0 and sin ( 7j — 7a ) = 0,

...(C),

or else

c °s(£, + ft 2 ) = 0 and cos ( 7l - 7a ) = o,
s * n Oft ~ /ft) = 0 and cos (/ft + /ft) = ().

or lastly

POLARIZED LIGHT FROM DIFFERENT SOURCES.

239

The first case gives /3 2 = /3 1? 7 1 -y 2 = a 1 -a 2 = 0 or ±180°, so
that the two streams into which it was proposed to resolve the
first are of the same nature. The second case gives /3 2 = 90° — f3 lt
Yi '~72 = a i~” a * 2 := ± 90°, which shews that the two streams are
identical in their nature, only the first and second principal
planes of the first of these streams are accounted respectively
the second and first of the second stream. The third case gives
/3 2 = /?! = ±45°, so that the two streams are circularly polarized
and of the same kind, which is a particular instance of the first
case.

Hence, universally, a stream of elliptically polarized light may
be resolved into two streams of elliptically polarized light in which
the polarizations are of any kind that we please, but different from
one another.

Substituting for q 2 , &c. their values in (7), and replacing y v y 2
by a x — a, a 2 — a, for which they had been temporarily written, we
find

{sin(/3 2 ~/3)cos(a 2 ~0 \

-J-I cos (/3 2 + /3 ) sin (a 2 — a )}" 1 G x

= {sin (/3 — /3 X ) cos (a — a x )

- J -1 cos (/3 + /3 X ) sin (a - aJ}" 1 0 2

= {sin(/3 2 -/3 1 )cos(a 2 -a 1 )

- J - 1 cos (/3 2 -f /3 X ) sin (a 2 - a^}" 1 Q )

...( 8 ).

3. Among these various modes of resolution there is one
which possesses several peculiar properties, any one of which
might serve to define it. Let us in the first place examine
under what circumstances the intensity of the stream made up
of the two components is independent of any retardation which
the phase of vibration of one component may have undergone
relatively to the phase of vibration of the other previously to
the recomposition.

For this purpose there is evidently no occasion to consider the
manner in which a v b x) e 1? a 2 , b 2 , e 2 are made up of a, b, e, but we
may start with the components. Let p v p 2 be the retardations
of phase which take place before recomposition, and resolve the
disturbances along the axes of w, y. We shall have for the
resolved parts

240 ON THE COMPOSITION AND RESOLUTION OF STREAMS OF

as = 8 {a t cos a x sin (<j> 4 - ^ - p x ) — ^ sin ^ cos (<j> + € 1 ~
y = 8 {a 4 sin a x sin (<jf> 4 e A - p t ) 4 cos ^ cos ($ 4 e x - p^},

where $ denotes the sum of the expression written down and
that formed from it by replacing the suffix 1 by 2 . To form
the expression for the intensity, or rather what would be the
intensity if the quantities c and e were absolutely constant, not
merely constant for a great number of successive undulations,
we must develope the expressions for os and y so as to contain
the sine and cosine of <£>-{-/c, and take the sum of the squares
of the coefficients, k is here a constant quantity which may be
chosen at pleasure, and which it will be convenient to take equal
to — p v If I be the intensity, in the sense above explained,
or as it may be called the temporary intensity, we find, putting
B for e 2 -p 2 -€ 1 + p 1 ,

I = [a x cos ol x 4 a 2 cos a 2 cos 8 4 - b 2 sin a 2 sin 8} 2

4 - {— b t sin a x 4 a 2 cos a 2 sin 8 — b 2 sin a 2 cos 8} 2

4 [a x sin ol x -I- a 2 sin a 2 cos 8 — b 2 cos a 2 sin 8} 2

4* cos 4 a 2 sin a 2 sin 8 4 b 2 cos a 2 cos 8} 2

= a* 4 b* 4 a 2 4 b 2 4 2 (a^ 4 bj) 2 ) cos (a 2 - a x ) cos 8

4 2 (afi 2 4 a 2 b t ) sin (a 2 - a t ) sin 8 .

On putting for a, b their values c cos /3, c sin ft, this expression
becomes

/ = c 2 4 c 2 4 2c,c 9 {cos (a fi - a.) cos (&- &) cos 8

4 sin (a 2 - oq) sin (/3 2 4 /3 X ) sin 8 } ... (9).

In order that I may be independent of the difference of phase
p 2 — /?i, and therefore of 8, we must have either

cos (a 2 - a x ) = 0, sin (£ 2 4 ft) = 0,.(10),

or sin (oc 2 - oq) = 0, cos (/3 2 — /3J = 0,.(11).

The equations (10) give a 2 — <aq = + 90°, /S 2 = — /3 1? so that the
ellipses described in the case of the two streams are similar, their
major axes perpendicular to each other, and the direction of revo¬
lution in the one stream contrary to that in the other. It will
be easily seen that the equations (11) differ from (10) only in
this, that what are regarded as the first and second principal
planes of the second stream when equations (10) are satisfied,
are accounted respectively the second and first when (11) are
satisfied.

POLARIZED LIGHT FROM DIFFERENT SOURCES.

241

Two streams thus related may be said to be oppositely polar¬
ized. Two streams of plane-polarized light in which the planes
of polarization are at right angles to each other, and two streams
of circularly polarized light, one right-handed and the other left-
handed, are particular cases of streams oppositely polarized.

In the reasoning of this article, nothing depends upon the pre¬
cise relation between the two polarized streams and the original
stream. All that it is necessary to suppose is, that the two polar¬
ized streams came originally from the same polarized source, so
that the changes in epoch and intensity, that is, the changes in
the quantities e, c, are the same for the two streams. Nothing
depends upon the precise nature of these changes, which may be
either abrupt or continuous, but must be sufficiently infrequent
if abrupt, or sufficiently gradual if continuous, to allow of our
regarding c and e as constant for a great number of successive
undulations. Our results will apply just as well to the dis¬
turbance produced by the union of two neighbouring streams
coming originally from the same polarized source, but having
had their polarizations modified, as to that produced by the
union, after recomposition, of the components of a single polar¬
ized stream. Since the resulting intensity is independent of S,
it follows that two oppositely polarized streams coming originally
from the same polarized source are incapable of interfering, but
two streams polarized otherwise than oppositely necessarily inter¬
fere, to a greater or less degree, when the difference in their
retardation of phase is sufficiently small. Of course the inter¬
ference here spoken of means only that which is exhibited without
analyzation.

4. Two interfering streams may be said to interfere perfectly
when the fluctuations of intensity are the greatest that the differ¬
ence in the intensities of the interfering streams admits of, so
that in case of equality the minima are absolutely equal to zero.
Referring to (9), we see that in order that this may be the case
the maximum value of the coefficient of 2 cp 2 must be equal to 1.
Now the maximum value of A cos S + /3 sin S is V (A 2j r R 2 ), and
therefore we must have

cos 2 (ol 2 - Oj) cos 2 (/3 2 - /3j) + sin 2 (ot 2 - a x ) sin 2 (# 2 + /3 t ) = 1

= cos 2 (a a - a t ) -|- sin 2 (a 2 - a t ),
16

s. III.

242 ON THE COMPOSITION AND RESOLUTION OF STREAMS OF
whence,

cos 2 (a 2 - cq) sin 2 (/3 2 - ft) + sin 2 (a, - oq) cos 2 (£ 2 + /3 X ) = 0,

which leads to the very same conditions that have been already
discussed in Art. 2. Hence two polarized streams coming from
the same polarized source are capable of interfering perfectly
if the polarizations are the same, not at all if the polarizations
are opposite, and in intermediate cases of course in intermediate
degrees.

5. When a stream of polarized light is resolved into two
oppositely polarized streams, which are again compounded after
their phases have been differently altered, we have from (9),
taking account of (10) or (11),

+ . ( 12 ),

so that the intensity of the resultant is equal to the sum of the
intensities of the components, and is therefore constant, that is,
independent of p x — p 2 , and is accordingly equal to what it was at
first, when p x and p 2 were each equal to zero, that is, equal to the
intensity of the original stream.

It may be readily proved from the formulae (8) that it is only
in the case in which the polarizations of the two components of
the original polarized stream are opposite that the intensity of
the original stream, whatever be the nature of its polarization, is
equal to the sum of the intensities of the component streams.
For, changing the sign of V(~l) in these formulae, multiplying
the resulting equations, member for member, by the equations
(8), and observing that if G' be what G becomes, GG' = + h? = c\

we find

{sin 2 (/3 2 — )cos 2 (a 2 — a ) + cos 2 (/3 2 + /3 )sin 2 (a a — a )}" 1 c 1 2> j
= {sin 2 (/3 - £,) cos 2 (oq- a ) + cos 2 (/? + f} x ) sin 2 (a 1 ~ a )J" 1 c 2 2 i(18).

= {sin 2 (/3 S - /3,) cos 2 (a 2 - a,) + cos 2 (£ 2 + £,) sin 2 (a 2 - a,)} -1 c 2 j

In order that ttt (c 2 ) may be equal to m (c x 2 ) -f m (c 2 2 ), it is
necessary that c 2 be equal to c x -f c 2 2 , because, whatever fluctua¬
tions c x and c 2 may undergo in a moderate time, such as the
tenth part of a second, c x and c 2 are always proportional to c.
Hence the sum of the quantities whose reciprocals are the co¬
efficients of Cj 2 and c 2 2 must be equal to that whose reciprocal is
the coefficient of c 2 . Since this has to be true independently of

POLARIZED LIGHT FROM DIFFERENT SOURCES.

243

/3, let the quantities sin 2 (/3 2 — /3), &c. be replaced by sines and
cosines of multiple arcs, and let our equation be put under the
form

A + B cos 2/3 -f C sin 2/3 = 0.

Then A } B, 0 must be separately equal to zero, or

sin 2 (/3 2 — /3,) cos 2 (a 2 — oq) + cos 2 (/3 2 + /3 X ) sin 2 (a 2 - a,) = 1;
cos 2/3 2 cos 2 (a 2 — a) + cos 2/3 1 cos 2 (cq — a) = 0;
sin 2/3 2 + sin 2/3 1 = 0.

(14).

Replacing unity in the right-hand member of the first of these
equations by

cos 2 (a 2 - oq) + sin 2 (a 2 - oq),

we find

cos 2 (/3 2 — ySJ cos 2 (a 2 - oq) + sin 2 (/3 2 + /3J sin 2 (oq — oq) = 0;

whence /3 2 = — /3 1? a 2 =a 1 +90°, or else /3 2 and ^ differ by 90°,
and a 2 = oq, except in the particular case in which /3 1 = ± 45°,
when /3 2 = + 45° satisfies the equation independently of ct 2 . Hence
the streams must be polarized oppositely, a condition which may
always be expressed by

/S 2 =~/3 1 , a 2 =oq + 90°,

which equations satisfy the second and third of equations (14)
independently of a, as it might have been foreseen that they
would, since it has been already shewn that the condition (12)
is satisfied in the case of oppositely polarized streams. It now
appears that it is only in the case of such streams that this is
satisfied.

6. The properties of oppositely polarized pencils which have
been proved, render it in a high degree probable that it is a
general law that in a doubly refracting medium the two polarized
pencils transmitted in a given direction are oppositely polarized.
Were this not the case, the two pencils, polarized otherwise than
oppositely, into which a polarized pencil is resolved on entering
into the medium, would at emergence compound a pencil of which
the intensity would depend upon the retardations of phase of one
pencil relatively to the other, so that such a medium, when ex¬
amined with polarized light, ought to exhibit rings or colours
without the employment of an analyzer. It is here supposed

16-2

244 ON THE COMPOSITION AND RESOLUTION OF STREAMS OF

that the light enters the medium at the first surface, and leaves
it at the second, in a direction normal to the surface, or very
nearly so, and likewise that the refracting power of the medium
for the two pencils is not very different, so that the effect of re¬
flexion at the two surfaces may be disregarded, the only sensible
effect being to diminish the intensity of each of the two pencils
in the same proportion, without affecting its state of polariza¬
tion. These views would lead us to scrutinize very carefully any
experimental evidence brought forward which would lead to the
conclusion, that the two polarized pencils which a doubly re¬
fracting medium was capable of propagating in a given direction
were polarized otherwise than oppositely.

7. In ordinary doubly refracting crystals, whether uniaxal or
biaxal, and in doubly refracting liquids, such as syrop of sugar,
it is generally admitted that the two pencils transmitted in a
given direction are oppositely polarized. In the former case the
two pencils are polarized in rectangular planes, in the latter
they are circularly polarized, one being right-handed and the
other left-handed. The same is the case with quartz for pencils
transmitted in the direction of the axis; but in following out
the researches in which he so successfully connected the pheno¬
mena, identical with those of a doubly refracting liquid, which
quartz exhibits in the direction of the axis, with the phenomena
which it exhibits as a uniaxal crystal, Mr Airy met with an ex¬
perimental result which seemed to shew that while the ellipses
which characterize the two streams transmitted in a given direc¬
tion, oblique to the axis, have their major axes situated, one in
a principal plane, and the other perpendicular to the principal
plane, and the directions of revolution are opposite, the eccen¬
tricities of the ellipses, though nearly, are not quite equal*.
This conclusion depended upon the result of certain experiments
made by means of a Fresnel's rhomb. The nature of the ex¬
periments seemed to eliminate the effect of an error in the
rhomb, that is, a deviation from 90° in the retardation of phase
which it produced in a pencil polarized in the plane of reflexion
relatively to a pencil polarized in a plane perpendicular to the
former. The effect of a possible index error in the plane of
reflexion seemed also to be eliminated; and the quantities on

* Cambridge Philosophical Transactions , VoL iv. p. 204.

POLARIZED LIGHT FROM DIFFERENT SOURCES.

245

which the results depended, though small, seemed to be beyond
mere errors of pointing. Being impressed however with a strong
conviction that the result depended in some way on the mode of
observation, I was led to scrutinize the different steps of the
process, and it occurred to me that the apparent inequality of
eccentricities was probably due to defects of annealing in the
rhomb. It is next to impossible to procure a piece of glass of
such a size free from defects of that nature, for which reason I
believe that even a good Fresnels rhomb is not to be trusted
for minute quantities, except in the case of merely differential
observations.

The same views lead to the conclusion that the two pencils
transmitted through magnetized glass in a direction oblique
to the lines of magnetic force are oppositely polarized. Some
theoretical investigations in which I was engaged some time ago
led me to the result that these polarized streams are circularly
polarized, as well as those transmitted along the line of magnetic
force, and that the difference between the wave-velocities varies
as the cosine of the inclination of the wave-normal to the line
of magnetic force. I hope at some future time to bring these
researches before the notice of this Society.

8. After these preliminary investigations respecting the na¬
ture of opposite polarization, which indeed contain, it is probable,
but little that has not already occurred to persons who have
studied the subject, it is time to come to the more immediate
object of this paper, which relates to the combination of inde¬
pendent streams. But first it will be convenient to state ex¬
plicitly a principle which is generally recognized.

When any number of polarized streams from different sources
mix together, after having been variously modified by reflexion,
refraction, transmission through doubly refracting xnedia, tourma¬
lines, &e., the intensity of the mixture is equal to the sum of the
intensities due to the separate streams.

The reason of this law may be easily seen. The components
whereby the disturbance due to any one stream is originally ex¬
pressed have to be resolved, their components resolved again, and
so on • and of these partial disturbances the phases of vibration
have to be altered by quantities independent of the time, and the

246 ON THE COMPOSITION AND RESOLUTION OF STREAMS OF

coefficients in some cases diminished in given ratios, and in some
cases suppressed altogether. Each stream has to be treated in
a similar way. The final disturbance being resolved in any two
rectangular directions, each component must be put under the
form U cos <£ + V sin <£, and the sum of the squares of U and V
must be taken to form the expression for the temporary in¬
tensity. All the quantities such as U and V will evidently be
linear functions of c cos e, c sin e, d cos e', d sin e', &c., where
c, d ... and e, e'... refer to the different streams, so that U for
instance will be of the form

Ac cos e 4* Be sin e + A V cos e + B'd sin d + ...

where A , B } A', B' ... are independent of the time. The tem¬
porary intensity will involve U 2 , but the actual intensity will
involve m(!!7 2 ), or

ttt2 (Ac cos e 4- Be sin e) 2 + 2m2 {(Ac cos e + Be sin e)

( A f d cos e' 4- B'c' sin e')}.

Now the products such as cos e cos e } cos e sin e', &c. will
have a mean value zero, since the changes in e and those in
€ have no relation to each other, and therefore the expression
for nt ( U 2 ) becomes

m2 (Ac cos e +Be sin e) 2 , or 2ui (Ac cose + Be sine) 2 ,

that is, the sum of the quantities by which it would be expressed
were the different streams taken separately.

Two streams which come from different sources, or which,
though in strictness they come from the same source, are such
that the changes of epoch and intensity in the one have no
relation to the changes of epoch and intensity in the other, may
be called independent

9. Suppose that there are any number of independent polar¬
ized streams mixing together; let the mixture be resolved in any
manner into two oppositely polarized streams, and let us examine
the intensity of each.

Let us take one stream first. The intensities of its compo¬
nents are given by the formula (13), which become somewhat
simpler in the case of opposite polarizations, since (3 2 = — (3 V and
^=90° + ^. Hence

c i — [sin 2 (/?,. +@) sin 2 (a t - a) + cos 2 (/^ — f3) cos 2 (o^ - a)} d ;

POLARIZED LIGHT FROM DIFFERENT SOURCES.

247

whence we find

m (c x ) = i {1 + sin 2/3 1 sin 2/3 + cos 2a t cos 2a cos 2/Sj cos 2 ft

+ sin 2a t sin 2a cos 2 /3 X cos 2/3} nt (c 2 ) ... (15).

There will be no occasion to write down the value of UT (c a 2 ),
since c t may be taken to refer to either component.

Let us pass now to the consideration of a group consisting of
any number of independent polarized streams. Let

£m (c 2 ) = A, 2 sin 2/3XXX (c 2 ) = B, 2 cos 2a cos 2/3ttt (c 2 ) = C
2 sin 2a cos 2/3W (c 2 ) = D,

and let c x now refer to one of the components of the whole group;
then

2m (c x ) = A + B sin 2/3 t + C cos 2oq cos 2/3 x + D sin 2a t cos 2/3 x .. .(17).

It follows that if there are two groups of independent polar¬
ized streams which are such as to give the same values to each
of the four quantities A, B , C } D defined by (16), if the groups
be resolved in any manner whatsoever, which is the same for
both, into two oppositely polarized streams, the intensities of the
components of the one group will be respectively equal to the
intensities of the components of the other group. Conversely, if
two groups of oppositely polarized streams are such that when
they are resolved in any manner, the same for both, into two
oppositely polarized streams, the intensities of the components of
the one group are respectively equal to the intensities of the
components of the other group, the quantities A, B, C, D must
be the same for the two groups. For, if we take accented letters
to refer to the second group, the second member of equation (17),
and the expression thence derived by accenting A, B , G } D , must
be equal independently of a i and /3,, which requires that A\ B\
G'y D' be respectively equal to A , B , 0, D.

Definition. Two such groups will be said to be equivalent .

10. The theoretical definition of equivalence which has just
been given agrees completely with the experimental tests of
equivalence. One of the most ready as well as delicate modes
of detecting minute traces of polarization, and at the same time
determining qualitatively the nature of the polarization, consists
in viewing the light to be examined through a plate of calcareous

248 ON THE COMPOSITION AND RESOLUTION OF STREAMS OF

spar or other crystal, cut for shewing rings, followed by a Nicol’s
prism. The plane-polarized pencils respectively stopped by and
transmitted through the Nicol’s prism consisted, on entering the
crystal, of pencils elliptically polarized in opposite ways; and
the nature of this elliptic polarization changes in every possible
manner from one point to another of the field of view. If these,
two streams of light be equivalent according to the definition
given in the preceding article, they will present exactly the
same appearance on being viewed through a crystal followed by
a Nicofs prism or other analyzer.

11. Theorem. Let a polarized stream be resolved into two
oppositely polarized streams; let the phase of vibration of one
of the streams be altered by a given quantity relatively to that
of the other, and let the streams be then compounded. If the
polarization of the original stream be now changed to its oppo¬
site, the polarization of the final stream will also be changed to
its opposite.

The straightforward mode of demonstrating this theorem, by
making use of the general expressions, would lead to laborious
analytical processes, which are wholly unnecessary. For the
formulae which determine the components of a given stream are
expressed by simple equations, so that the results are unique, and
accordingly whenever we can foresee what the result will be, it is
sufficient to shew that the formulae themselves, or the geometrical
conditions of which the formulae are merely the expressions, are
satisfied.

For shortness 3 sake call the original stream X, and its com¬
ponents 0, E. Let p be the given quantity, positive or nega¬
tive, by which the phase of vibration of 0 is retarded relatively
to that of E. Let o, e denote the streams 0, E after the changes
of phase, and Y the stream resulting from their reunion. Con¬
ceive now all the vibrations with which we arc concerned to be
turned in azimuth through 90°. This will not affect the geo¬
metrical relations connecting components and resultants. Let
X', O', E', o', e, Y' be the streams which X, 0, E, o, e, Y thus
become. The streams O', E' are evidently polarized in the same
manner respectively as E, 0, except that right-handed is changed
into left-handed, and vice versa; and in passing from O' to d the
phase is retarded by p. Now conceive the direction of motion of

POLARIZED LIGHT FROM DIFFERENT SOURCES.

249

a given particle reversed, the motions of all other particles being
derived from that of the first according to the general law of wave
propagation. The relations between components and resultants
will evidently not be violated; and if X", 0", E", o", e n , F" denote
the streams into which X', 0\ E\ o', e’, Y' are thus changed, it
is evident that the polarizations of X", 0", E", o", e", Y" are
respectively opposite to those of X, 0, E, o, e, Y. But on ac¬
count of the reversion in direction of motion it is plain that there
is reversion as regards acceleration and retardation of phase, so
that in passing from the pair 0", E" to the pair o", e" the phase
of 0" is accelerated by p relatively to the phase of E".

Hence the stream X", polarized oppositely to X, is resolved
into two E", 0", polarized in the same manner respectively as
0, E, which are recompounded after the phase of the one polarized
in the same manner as 0 has been retarded by p relatively to the
phase of the other, and the result is Y", a stream polarized in a
manner opposite to F, which proves the theorem.

12. This theorem may bo applied to the case of light
transmitted through a slice of a doubly refracting crystal, and
shews that two streams going in oppositely polarized come out
oppositely polarized. Also, since nothing in the demonstration
depends upon the order in which the decompositions and recom¬
positions take place, it is immaterial whether X , X" denote a
pair of oppositely polarized incident streams, which give rise to
the emergent streams F, F", or X , X" denote a pair of oppo¬
sitely polarized emergent streams, which came from the incident
streams F, Y'. A particular case of this theorem was assumed
in the preceding article, when it was stated that the pencils,
polarized in perpendicular planes, which on coming out of a
crystal of calcareous spar are respectively stopped by and trans¬
mitted through a Nicol’s prism, went into the crystal oppositely
polarized.

The theorem will evidently be true of a train consisting of
any number of crystalline plates, each possessing the property
of resolving the incident light into two oppositely polarized
streams, which arc propagated within the medium with different
velocities. For two oppositely polarized streams incident on the
first plate give rise to two emergent streams which fall oppo¬
sitely polarized on the second plate, and so on. Since the number

250 ON THE COMPOSITION AND RESOLUTION OF STREAMS OF

of plates may be supposed to increase and their thickness to de¬
crease indefinitely, while at the same time the steps by which
the doubly refracting nature of the plates alters from one to
another become separately insensible, the theorem will be true
if the whole train, or part of it, consist of a substance of which
the doubly refracting nature alters continuously, as for example
a piece of strained or unannealed glass. If, however, the train
contain a member which performs a partial analysis of the light,
as for example a plate of smoky quartz, or a plate of glass in¬
clined to the incident light at a considerable angle, it will no
longer be true that two pencils going in oppositely polarized will
come out oppositely polarized.

13. Theorem. If two equivalent groups of polarized streams
be resolved in any manner, which is the same for both, into two
oppositely polarized groups, and these be recombined after the
phase of one of the components has been retarded by a given
quantity relatively to that of the other, the two groups of resultant
streams will be equivalent.

Let the groups of resultants be each resolved in any manner
into two oppositely polarized streams, and call these 0, E. By
Art. 11, if O', E' be the streams which furnish 0, E respectively,
O', E' are oppositely polarized. Now by Art. 3, the intensities
of 0, E are the same respectively as those of O', E'\ but these
are the same for the one group as for the other, by the defini¬
tion of equivalence. Therefore the intensities of 0, E are the
same for the one group as for the other; but 0, E are any two
oppositely polarized components of the resultant groups; therefore
these groups are equivalent.

Hence, if two equivalent groups be transmitted through a
crystalline plate, the emergent groups will be equivalent; and
by the same reasoning as in Art. 12 the theorem may be ex¬
tended to an optical train consisting of any number of crystalline
plates, pieces of unannealed glass, &c.

14. Theorem. If two equivalent groups be resolved in any
manner, the same for both, into two polarized streams, the in¬
tensities of the components of the one group will be respectively
equal to the intensities of the components of the other group.

The proof of this theorem is very easy. It is sufficient to
treat the general expressions (13) exactly as in Art. 9, only that

POLARIZED LIGHT FROM DIFFERENT SOURCES.

251

no relations are to be introduced between a 2 , /3 2 , and a 1}
Since the coefficient of c 2 in (13) is constant, if we consider as
variable such quantities only as may change in passing from the
one group to the other, it will be easily seen that the intensity
of either component depends on the same four constants A, B,

' C } B as before; and since these have the same values for equi¬
valent groups, it follows that in the most general mode of reso¬
lution the components of any two such groups have the same
intensities respectively.

15. Theorem. If two equivalent groups be each resolved
in the same manner into two oppositely polarized streams, and
these be recombined after their vibrations have been diminished in
two different given ratios, the resultant groups will be equivalent.

Let each of the resultant groups be resolved into two oppo¬
sitely polarized streams 0, E, and let O', E' denote the streams
which furnished 0, E respectively. It is easily seen that the
streams O', E' are both polarized, though not in general oppo¬
sitely. Were there any occasion to determine the nature of the
polarizations, it might easily be done by following a process the
reverse of that by which the modified are deduced from the
original groups. Thus, if it were required to determine the
polarization of O', we should resolve 0 into streams oppositely
polarized in the manner originally given, augment the vibra¬
tions in ratios the inverse of those in which they are actually
diminished, and recombine the streams so obtained. For our
present purpose, however, it is sufficient to observe that the
polarizations of O', E f depend only on the mode of resolution,
and not on the nature of the original group, and that therefore
they arc the same for the one group as for the other. The
intensity of 0' is not the same as that of 0, as in the case con¬
sidered in Art. 13, but bears to it a ratio depending only on the
mode of resolution, and therefore the same for each of the two
original groups. The same is true of the intensity of E' com¬
pared with that of E. Now by Art. 14 the intensities of O', E
arc the same for the two original groups, and the intensities of
0, E bear to those of O', E* respectively, ratios the same for the
two groups: therefore the intensities of 0, E arc the same for the
two final groups; but 0, E are any two oppositely polarized com-
■ ponents of these groups ; therefore these groups are equivalent.

252 ON THE COMPOSITION AND RESOLUTION OF STREAMS OF

16. It follows from this theorem that no partial analysis of
light, such, for example, as would he produced by reflexion from
the surface of glass or metal, or by transmission through a doubly
absorbing medium, can from equivalent groups produce groups
which are not equivalent to each other; and we have seen
already that this cannot be done by means of the alteration of
phase accompanying double refraction. It follows, therefore, that
equivalent groups are optically undistinguishable.

In proving this property of equivalent groups, it has been
supposed that the polarizations of the two streams into which
any group was resolved were opposite, such being the case in
nature. But did a medium exist such that the two streams of
light which it transmitted independently were polarized other¬
wise than oppositely, it would still not enable us to distinguish
between equivalent groups.

17. The experimental definition of common light is, light
which is incapable of exhibiting rings of any kind when ex¬
amined by a crystal of Iceland spar and an analyzer, or by some
equivalent combination. Consequently, a group of independent
polarized streams will together be equivalent to common light
when, on being resolved in any manner into two oppositely
polarized pencils, the intensities of the two are the same, and
of course equal to half that of the original group. Accordingly,
in order that the group should be equivalent to common light,
it is necessary and sufficient that the constants B , C } D should
vanish.

18. Let us now sec under what circumstances two inde¬
pendent streams of polarized light can together be equivalent
to common light.

Let a, ft refer to the first, and a, ft' to the second stream, and
let the intensities of the two streams be as 1 to n\ then wc get
from the formulas (16)

sin 2ft 4- n sin 2ft' = 0 ;
cos 2a cos 2ft 4- n cos 2a! cos 2 ft! = 0 ;
sin 2a cos 2ft 4- n sin 2a cos 2ft' = 0.

Transposing, squaring, and adding, we find u 2 = 1, and there¬
fore ?i=l, since n is essentially positive. Since ft and ft' are

POLARIZED LIGHT FROM DIFFERENT SOURCES.

253

supposed not to lie beyond the limits —90° and 4-90°, we get
from the first equation ft' = — ft } or ft' = + 90° — ft, + or — ac¬
cording as ft is positive or negative. Now it is plain that any
one solution must be expressed analytically in two ways, in which
the values of ft' are complementary, and the values of a differ
by 90°, since either principal axis of the ellipse belonging to the
second stream may be that whose azimuth is a'. Accordingly,
we may reject the second solution as being nothing more
than the first expressed in a different way, and may therefore
suppose ft' — — ft. The second and third equations then give
cos 2a' = — cos 2a, sin 2 a = — sin 2a, and therefore a and a' differ
by 90°. The equations are indeed satisfied by ft = — ft'=± 45°,
but this solution is only a particular case of the former.

It follows therefore that common light is equivalent to any
two independent oppositely polarized streams of half the in¬
tensity; and no two independent polarized streams can together
be equivalent to common light, unless they be polarized oppo¬
sitely, and have their intensities equal.

19. We have seen that the nature of the mixture of a given
group of independent polarized streams is determined by the
values of the four constants A, B } C, D. Consider now the
mixture of a stream of common light having an intensity and
a stream, independent of the former, consisting of elliptically
polarized light having an intensity J' } and having a for the
azimuth of its plane of maximum polarization, and tan ft' for
the ratio of the axes of the ellipse which characterizes it.

By the preceding article, the stream of common light is equi¬
valent to two independent streams, plane-polarized in azimuths 0°
and 90°, having each an intensity equal to \J. Hence, applying
the formulae (16) and (17) to the mixture, we have

2m (Cj) 2 = J + /' -f J 7 sin 2ft' sin 2ft t + J' cos 2a' cos 2/3'cos 2a x cos 2ft x

+ J' sin 2a cos 2ft' sin 2a x cos 2^ ;

and this mixture will be equivalent to the original group of polar¬
ized streams, provided

J + J' = Ai
J' cos 2a' cos 2ft' ~ C ;

J' sin 2ft' = B ; j
J' sin 2a' cos 2ft' = D. J

254 ON THE COMPOSITION AND RESOLUTION OF STREAMS OF

These equations give

JWGB 2 +C 2 + D 2 );

JB

sm 2 ^ = V(F+C ! + F) ;

J = A-^/(B‘ 2 +G i + D 2 );

tan 2a' =

D

O'

...(19).

These formulae can always be satisfied, and therefore it is
always possible to represent the given group by a stream of
common light combined with a stream of elliptically polarized
light independent of the former. Moreover, there is only one
way in which the group can be so represented. For, though the
third of equations (19) gives two values for /3' complementary to
each other, these values, as before explained, lead only to two
ways of expressing the same result. If we choose that value of
ft which is numerically the smaller, then among the different
values of a', differing by 90°, which satisfy the fourth equation, we
must choose one which gives to cos 2 a the same sign as G

20. Let us now apply the principles and formulae which have
just been established to a few examples. And first let us take
one of the fundamental experiments by which MM. Arago and
Fresnel established the laws of interference of polarized light, or
rather an analogous experiment mentioned by Sir John Herschel.
The experiment selected is the following.

Two neighbouring pencils of common light from the same
source are made to form fringes of interference. A tourmaline,
carefully worked to a uniform thickness, is cut in two, and its
halves interposed in the way of the two streams respectively. It
is found that when the planes of polarization of the two tour¬
malines are parallel the fringes are formed perfectly; but as one
of the tourmalines is turned round in azimuth the fringes become
fainter, and at last, when the planes of polarization become per¬
pendicular to each other, the fringes disappear.

Let the planes of polarization of the tourmalines be inclined
at an angle a, and let it be required to investigate an expression
for the intensity of the fringes. Since common light is equi¬
valent to two independent streams, of equal intensity, polarized
in opposite ways, let the original light be represented by two
independent streams, having each an intensity equal to unity,
polarized in planes respectively parallel and perpendicular to the
plane of polarization of the first tourmaline. If c, c be the co-

POLARIZED LIGHT FROM DIFFERENT SOURCES.

255

efficients of vibration in the two streams respectively, c cos a,
c' sin a will be the coefficients of the resolved parts in the direc¬
tion of the vibrations transmitted by the second tourmaline.
Hence we shall have, mixing together, two independent streams,
in one of which the temporary intensity fluctuates between the
limits (c±c cos 2 a) 2 +- (c cos a sin a ) 2 as the interval of retarda¬
tion changes in passing from one point to another in the field
of view. The temporary intensity of the other stream being
{o' sin a) 2 , the intensity at different points will fluctuate between
the limits

nt (c 2 ± 2 c 2 cos 2 a + c 2 cos 2 a) + m (c' sin a) 2 .

It is needless to take account of the absorption which takes
place even on the pencils which the tourmalines do transmit,
because it affects both pencils equally. Since m (c 2 ) = Rl (c /2 ) = 1,
we have for the limits of fluctuation of the intensity 2 + 2 cos 2 a.
When a = 0 these limits become 4 and 0, and the interference is
perfect. When a = 90° the limits coalesce, becoming each equal
to 2, and there are no fringes. As a increases from 0 to 90°, the
superior limit continually decreases, and the inferior increases, and
consequently the fringes become fainter and fainter.

21. It is a well-known law of interference that if two rays
of common light from the same source be polarized in rect¬
angular planes, and afterwards be brought to the same plane of
polarization, or in other words, analyzed so as to retain only
light polarized in a particular plane, they will not interfere, but
if the light be primitively polarized in one plane they will inter¬
fere. This law seems to have presented a difficulty to some,
because, it would be argued, the most general kind of vibra¬
tions are elliptic, so that wc must suppose the vibrations of the
ether in the case of common light to be of this kind; and yet
the phenomena of interference are exhibited perfectly well if the
light be at first clliptically polarized instead of plane-polarized.
For my own part, I never could see the difficulty, but on the
other hand it seems to me that it would be an immense diffi¬
culty were the law anything else than what it is. For, if we
consider the rectangular components of the vibrations which
make up common light, these components being measured along
any two rectangular axes perpendicular to the ray, we must
suppose them to be independent of each other, or at least to

256 ON THE COMPOSITION AND RESOLUTION OF STREAMS OF

have no fixed relation to each other so far as regards the
changes in the mode of vibration, which we must suppose to
be taking place continually, though slowly, it may be, in com¬
parison with the time of a luminous vibration. To suppose
otherwise would be contrary to the idea of common light, in
which it is implied that on the average whatever we can say
of one plane passing through the ray, we can say of another:
whatever we can say of the direction one way round we can say
of the other way round.

At the end of his excellent Tract on the Undulatory Theory,
Mr Airy has shewn how the simple supposition of the existence,
in common light, of successive series of undulations, in which
the vibrations of one series have no relation to those of another,
would account at the same time for the interference of common
light and the non-interference of the pencils, polarized in rect¬
angular planes, into which common light may be conceived to
be decomposed. But he has, I think, introduced a gratuitous
difficulty into the subject, by asserting that it is necessary to
suppose the transition from one series into another to be abrupt,
and that a gradual change in the nature of the vibrations is
inadmissible. This assertion, which seems to have led others to
conceive that there was here a difficulty with which the undu¬
latory theory had to contend, seems to have resulted from an
investigation from which it appeared that common light could
not be represented by an indefinite series of elliptic vibrations,
in which the major axis of the ellipse was supposed to revolve
uniformly, rapidly, with regard to the duration of impressions on
the retina, though slowly with regard to the time of a luminous
vibration. I have elsewhere pointed out on what grounds I con¬
ceive that the instance of the revolving ellipse is not a case in
point, namely, that it is not a fair representation of common
light, because it gives a preponderance on the average to one
direction of revolution over the opposite, which is contrary to
the idea of common light*. Let us now apply the general
formulae (16) and (17) to this case.

Let c cos /3, c sin /3 be the semi-axes of the ellipse, a the
azimuth of the first axis at a given time. So far as relates to

* Report of the meeting of the British Association at Swansea in 1848. Trans¬
actions of the Sections, p. 5.

POLARIZED LIGHT FROM DIFFERENT SOURCES. 257

the stream for which the azimuth of the first axis lies between
a and a + da, we have m (c 2 ) = (27r)' 1 c 2 cZa; and in the application
of the formulae (16) the summation, to which X refers, will of
course pass into an integration. We have therefore

A =

c 2

to

»

ii

= c 2 ;

2tt

Jo

C 2

[2ir

B =

2tt

sin 2/3

da~c 2 sin 2/3;

/ 0

0 =

0 ;

D = 0;

whence we get from the formulae (19), supposing ft positive,

</' = c 2 sin 2/3; / = c 2 (1 — sin 2$); /3'=45°;

while d remains indeterminate. Hence the mixture is equiva¬
lent, not to common light alone, but to a stream of common
light having an intensity equal to c 2 (1 — sin 2/3), combined with
a stream of circularly polarized light, independent of the former,
having an intensity equal to c 2 sin 2/3, and being of the same cha¬
racter as regards right-handed or left-handed as the original
stream would be were the ellipse stationary. The result of
supposing ft negative is here assumed as obvious.

22. Suppose that a polarizing prism and a mica plate, which
produce elliptic polarization, are made to revolve together with
great rapidity. The stream of light thus produced will be equi¬
valent to the former. The only difference is that in the former
case c was supposed constant, whereas in the case of actual ex¬
periment it will be subject to the fluctuations mentioned at the
beginning of this paper; but the mean values represented by tit will
not be affected when these fluctuations are taken into account,
and therefore the same formulae will continue to apply. Hence,
if the polarization be circular the rotation will make no differ¬
ence; if it be plane, the light will appear completely depolarized;
in intermediate cases the result will be intermediate, and the
light will be equivalent to a mixture of common light and cir¬
cularly polarized light. The reader may compare these conclu¬
sions of theory with some experiments by Professor Dove*.

23. As a last example, let light polarized by transmission
through a Nicol’s prism be transmitted through a second Nicol’s

See Philosophical Magazine , Vol. xxx. (1847) p. 465.

s. iir.

17

258

ON THE COMPOSITION AND RESOLUTION, ETC.

prism, which is made to revolve uniformly and rapidly while the
first remains fixed.

Let a be the azimuth of the plane of polarization of the second
Nicol’s prism, measured from that of the first, c the coefficient of
vibration in the stream transmitted through the first prism. The
stream passing through the second prism is made up of an infinite
number of independent streams such as that whose intensity is
( 27 T)" 1 cos 2 adoc multiplied by the mean value of c 2 . Hence we have
from the formulae (16)

3 = 0; C = Jm(c 2 ); 3 = 0;

whence, taking the intensity of the original stream as unity, we
have

3 = / = h /3' = 0; a' = 0;

or the light is equivalent to a mixture of common light having
an intensity and light independent of the former, of the same
intensity, polarized at an azimuth zero. This result may be com¬
pared with one of Professor Dove’s experiments.

If a rotating crystalline plate, cut from a doubly refracting
crystal in a direction not nearly coinciding with the optic axis
or one of the optic axes, and not sufficiently thin to exhibit
colours in polarized light, be substituted for the rotating Nicol’s
prism, since the plate is too thick to allow of the exhibition of
any phenomena depending on the interference of the oppositely
polarized pencils, the effect will be just the same as in the case
of the Nicol’s prism, only the intensity of each stream will be
doubled.

In applying the formulae of this paper to experiments in
which one part of an optical train is made to revolve rapidly, it
must be understood that the other parts of the train are at rest,
or at least do not revolve with a velocity nearly equal to the
former. Otherwise, particular phenomena will be exhibited de¬
pending on the simultaneous movements of two or more parts
of the train, as appeared in Professor Dove’s experiments; and
in the calculation of these phenomena it will not be allowable to
substitute for the stream of light emerging from the polarizer, or
first revolving piece whatever it be, the streams of common and
elliptically polarized light to which, for general purposes, it is
equivalent.

[From the Proceedings of the Royal Society , Vol. vi., p. 195.]

Abstract of a paper “On the Change of Refrangibility

of Light*.”

[Read May 27, 1852.]

The author was led into the researches detailed in this paper
by considering a very singular phenomenon which Sir John
Herschel had discovered in the case of a weak solution of sulphate
of quinine, and various other salts of the same alkaloid. This
fluid appears colourless and transparent, like water, when viewed
by transmitted light, but exhibits in certain aspects a peculiar
blue colour. Sir John Herschel found that when the fluid was
illuminated by a beam of ordinary daylight, the blue light was
produced only throughout a very thin stratum of fluid adjacent to
the surface by which the light entered. It was unpolarized. It
passed freely through many inches of the fluid. The incident
beam, after having passed through the stratum from which the
blue light came, was not sensibly enfeebled nor coloured, but yet
it had lost the power of producing the usual blue colour when
admitted into a solution of sulphate of quinine. A beam of light
modified in this mysterious manner was called by Sir John Herschel
epipolized.

Several years before Sir David Brewster had discovered in the
case of an alcoholic solution of the green colouring matter of
leaves a very remarkable phenomenon, which he has designated

* [As the paper in question is of considerable length, and contains a good deal
of detail, it seemed desirable in this case to reprint the abstract as well as the
paper itself. In the abstract the reader will find in short compass the more
important of the results contained in the paper.]

17—2

260

ABSTRACT OF A PAPER

as internal dispersion. On admitting into this fluid a beam of
sunlight condensed by a lens, he was surprised by finding the path
of the rays within the fluid marked by a bright light of a blood-
red colour, strangely contrasting with the beautiful green of the
fluid itself when seen in moderate thickness. Sir David afterwards
observed the same phenomenon in various vegetable solutions and
essential oils, and in some solids. He conceived it to be due to
coloured particles held in suspension. But there was one circum¬
stance attending the phenomenon which seemed very difficult of
explanation on such a supposition, namely, that the whole or a
great part of the dispersed beam was unpolarized, whereas a beam
reflected from suspended particles might be expected to be polar¬
ized by reflexion. And such was, in fact, the case with those
beams which were plainly due to nothing but particles held in
suspension. From the general identity of the circumstances
attending the two phenomena, Sir David Brewster was led to
conclude that epipolic was merely a particular case of internal
dispersion, peculiar only in this respect, that the rays capable of
dispersion were dispersed with unusual rapidity. But what rays
they were which were capable of affecting a solution of sulphate
of quinine, why the active rays were so quickly used up, while the
dispersed rays which they produced passed freely through the
fluid, why the transmitted light when subjected to prismatic
analysis showed no deficiencies in those regions to which, with
respect to refrangibility, the dispersed rays chiefly belonged, were
questions to which the answers appeared to be involved in as
much mystery as ever.

After having repeated some of the experiments of Sir David
Brewster and Sir John Herschel, the author could not fail to take
a most lively interest in the phenomenon. The firm conviction
which he felt that two portions of light were not distinguishable
as to their nature otherwise than by refrangibility and state of
polarization, left him but few hypotheses to choose between,
respecting the explanation of the phenomenon. In fact, having
regarded it at first as an axiom that dispersed light of any
particular refrangibility could only have arisen from light of the
same refrangibility contained in the incident beam, he was led by
necessity to adopt hypotheses of so artificial a character as to
render them wholly improbable. He was thus compelled to adopt
the other alternative, namely, to suppose that in the process of

ON THE CHANGE OF REFRANGIJBILIT Y OF LIGHT.

261

internal dispersion the refrangibility of light had been changed.
Startling as such a supposition might appear at first sight, the
ease with which it accounted for the whole phenomenon was
such as already to produce a strong probability of its truth.
Accordingly the author determined to put this hypothesis to the
test of experiment.

The experiments soon placed the fact of a change of refrangi¬
bility beyond all doubt. It would exceed the limits of an abstract
like the present to describe the various experiments. It will be
sufficient to mention some of the more remarkable results.

A pure spectrum from sunlight having been formed in air in
the usual manner, a glass vessel containing a weak solution of
sulphate of quinine was placed in it. The rays belonging to the
greater part of the visible spectrum passed freely through the
fluid, just as if it had been water, being merely reflected here and
there from motes. But from a point about half-way between the
fixed lines Q and H to far beyond the extreme violet the incident
rays gave rise to light of a sky-blue colour, which emanated in all
directions from the portion of the fluid which was under the
influence of the incident rays. The anterior surface of the blue
space coincided of course with the inner surface of the vessel in
which the fluid was contained. The posterior surface marked the
distance to which the incident rays were able to penetrate before
they were absorbed. This distance was at first considerable,
greater than the diameter of the vessel, but it decreased with
great rapidity as the refrangibility of the incident rays increased,
so that from a little beyond the extreme violet to the end the
blue space was reduced to an excessively thin stratum adjacent to
the surface by which the incident rays entered. It appears
therefore that this fluid, which is so transparent with respect to
nearly the whole of the visible rays, is of an inky blackness with
respect to the invisible rays more refrangible than the extreme
violet. The fixed lines belonging to the violet and the invisible
region beyond were beautifully represented by dark planes inter¬
rupting the blue space. When the eye was properly placed, these
planes were of course projected into lines. The author has made
a sketch of these fixed lines, which accompanies the paper. They
may be readily identified with the fixed lines represented in
M. Becquerel’s map of the fixed lines of the chemical spectrum.
The last line seen in a solution of sulphate of quinine appears to

262

ABSTRACT OF A PAPER

be the line next beyond the last represented in M. Becquerel’s
map. Under very favourable circumstances two dusky bands
were seen still further on. Several circumstances led the author
to conclude that in all probability fixed lines might be readily
seen corresponding to still more refrangible rays, were it not
for the opacity of glass with respect to those rays of very high
refrangibility.

It is very easy to prove experimentally that the blue dispersed
light corresponding to any particular part of the incident spectrum
is not homogeneous light, having a refrangibility equal to that of
the incident rays, and rendered visible in consequence of its
complete isolation; but that it is in fact heterogeneous light,
consisting of rays extending over a wide range of refrangibility,
and not passing beyond the limits of refrangibility of the spectrum
visible under ordinary circumstances. To show this it is sufficient
to isolate a part of the incident spectrum, and view the narrow
beam of dispersed light which it produces through a prism held
to the eye.

In Sir David Brewster’s mode of observation, the beam of
light which was of the same nature as the blue light exhibited by
a solution of sulphate of quinine was necessarily mixed with the
beam due merely to reflexion from suspended particles; and in
the case of vegetable solutions, a beam of the latter kind almost
always exists, to a greater or less degree. But in the method of
observation employed by the author, to which he was led by the
discovery of the change of refrangibility, the two beams are
exhibited quite distinct from one another. The author proposes
to call the two kinds of internal dispersion just mentioned true
internal dispersion and false internal dispersion, the latter being
nothing more than the scattering of light which is produced by
suspended particles, and having, as is now perfectly plain, nothing
to do with the remarkable phenomenon of true internal dispersion.

Now that the nature of the latter phenomenon is better known,
it is of course possible to employ methods of observation by which
it may be detected even when only feebly exhibited. It proves to
be almost universal in vegetable solutions, that is, in solutions
made directly from various parts of vegetables. When vegetable
products are obtained in a state of isolation, their solutions some¬
times exhibit the phenomenon and sometimes do not, or at least
exhibit it so feebly that it is impossible to say whether what they

ON THE CHANGE OF EEFEANGIBIL1TY OF LIGHT.

263

do show may not be due to some impurity. Among fluids which
exhibit the phenomenon in a high degree, or according to the
author s expression are highly sensitive , may be mentioned a weak
decoction of the bark of the horse-chestnut, an alcoholic extract
from the seeds of the Datura stramonium , weak tincture of tur¬
meric, and a decoction of madder in a solution of alum. In these
cases the general character of the dispersion resembles that ex¬
hibited by a solution of sulphate of quinine, but the tint of the
dispersed light, and the part of the spectrum at which the dis¬
persion begins, are different in different cases. In the last fluid,
for example, the dispersion commences somewhere about the fixed
line D, and continues from thence onwards far beyond the extreme
violet. The dispersed light is yellow, or yellowish orange.

In the case of other fluids, however, some of them sensitive in
a very high degree, the mode in which light is dispersed internally
presents some very remarkable peculiarities. One of the most
singular examples occurs in the case of an alcoholic solution of the
green colouring matter of leaves. This fluid disperses a rich red
light. The dispersion commences abruptly about the fixed line B ,
and continues from thence onwards throughout the visible spectrum
and a little beyond. The dispersion is subject to fluctuations
intimately connected with the singular absorption bands exhibited
by this medium.

In order that a medium should be capable of changing the
refrangibility of light incident upon it, it is not necessary that the
medium should be a fluid, or a clear solid. Washed papers and
other opaque substances produce the same effect, but of course the
mode of observation must be changed. The author has observed
the change of refrangibility in various ways. It will be sufficient
to mention here that which was found most generally useful, which
he calls the method of observing by a linear spectrum. The method
is as follows.

A series of prisms and a lens are arranged in the usual manner
for forming a pure spectrum, but the slit by which the light enters,
instead of being parallel, is placed in a direction perpendicular to
the edges of the prisms. A linear spectrum is thus formed at the
focus of the lens, consisting of an infinite succession of images of
the slit arranged one after the other in the order of refrangibility,
and of course overlapping each other to a certain extent. The
substance to be examined is placed in the linear spectrum, and

264

ABSTRACT OF A PAPER

the line of light seen upon it is viewed through a prism held to
the eye. In this way it is found that almost all common organic
substances, such as wood, cork, paper, calico, bone, ivory, horn,
wool, quills, feathers, leather, the skin of the hand, the nails, are
sensitive in a greater or less degree. Organic substances which
are dark-coloured are frequently found to be insensible, but, on
the other hand, scarlet cloth and various other dyed articles are
highly sensitive. By means of a linear spectrum the peculiar
dispersion of a red light produced by chlorophyll, or some of its
modifications, may be observed not only in a solution, but in a
green leaf, or on a washed paper, or in a sea-weed.

The highly sensitive papers obtained by washing paper with
tincture of turmeric, or a solution of sulphate of quinine, or some
other highly sensitive medium, display their sensibility in a re¬
markable manner when they are examined in a linear spectrum.
In these cases, however, the paper produces a very striking effect
when merely held so as to receive a pure spectrum formed in the
usual manner, that is, with a slit parallel to the edges of the
prisms. Such a paper may be used as a screen for showing the
fixed lines belonging to the invisible rays, though they are not
thus shown quite so well as by using a solution. The extraordinary
prolongation of the spectrum seen when it is received on turmeric
paper has been already observed by Sir John Herschel, by whom
it w T as attributed to a peculiarity in the reflecting power of that
substance. Of course it now appears that the true explanation is
very different.

A high degree of sensibility appears to be rather rare among
inorganic compounds. Certain specimens of finer spar, as is
already known, give a copious internal dispersion of a deep blue
light; but this is plainly due to some foreign ingredient, the
nature of which is at present unknown. But there is one class of
inorganic compounds which are very remarkable for their sensibility,
namely, certain compounds of peroxide of uranium, including the
ornamental glass called canary glass, and the natural mineral
yellow uranite. In these compounds the dispersed light is found
on analysis to consist of bright bands arranged at regular intervals.
A very remarkable system of absorption bands is also found among
these compounds, which is plainly connected with the system of
bright bands seen in the spectrum of the dispersed light. The
connection between the absorption and internal dispersion ex^

ON THE CHANGE OE RE FRANGIBILITY OF LIGHT.

265

hibited by these compounds is very singular, and is of a totally
different nature from the connection which has been already
mentioned as occurring in solutions of the green colouring matter
of leaves.

There is one law relating to the change of refrangibility which
appears to be quite universal, namely, that the refrangibility of
light is always lowered by internal dispersion. The incident rays
being homogeneous, the dispersed light is found to be more or less
composite. Its colour depends simply on its refrangibility, having
no relation to the colour of the incident light, or to the circumstance
that the incident rays were visible or invisible. The dispersed
light appears to emanate in all directions, as if the solid or fluid
were self-luminous while under the influence of the incident rays.

The phenomenon of the change of refrangibility of light admits
of several important applications. In the first place it enables us
to determine instantaneously the transparency or opacity of a solid
or fluid with respect to the invisible rays more refrangible than
the violet, and that, not only for these rays as a whole, but for the
rays of each refrangibility in particular. For this purpose it is
sufficient to form a pure spectrum with sun-light as usual, em¬
ploying instead of a screen a vessel containing a decoction of the
bark of the horse-chestnut, or a slab of canary glass, or some other
highly sensitive medium, and then to interpose the medium to be
examined, which, if fluid, would have to be contained in a vessel
with parallel sides of glass. Glass itself ceases to be transparent
about the region corresponding to the end of the author's map, and
to carry on these experiments with respect to invisible rays of still
higher refrangibility would require the substitution of quartz for
glass. The reflecting power of a surface with respect to the
invisible rays may be examined in a similar manner.

The effect produced on sensitive media leads to interesting
information respecting the nature of various flames. Thus, for
example, it appears that the feeble flame of alcohol is extremely
brilliant with regard to invisible rays of very high refrangibility.
The flame of hydrogen appears to abound in invisible rays of still
higher refrangibility.

By means of the phenomena relating to the change of refrangi¬
bility, the independent existence of one or more sensitive substances
may frequently be observed in a mixture of various compounds.
In this way the phenomenon seems likely to prove of value in the

266 ON THE CHANGE OF REFll A NGI HI LITY* OF LIGHT.

separation of organic compounds. The phenomena sometimes also
afford curious evidence of chemical combinations; but this subject
cannot here be further dwelt upon.

The appearance which the rays from an electric spark produce
in a solution of sulphate of quinine, shows that the spark is very
rich in invisible rays of excessively high refrangibility, such as
would plainly put them far beyond the limits of the maps which
have hitherto been made of the fixed lines in the chemical part of
the solar spectrum. These rays are stopped by glass, but trans¬
mitted through quartz. These circumstances render it probable
that the phosphorogenic rays of an electric spark are nothing more
than rays of the same nature as those of light, but which are
invisible, and not only so, but of excessively high refrangibility.
If so, they ought to be stopped by a very small quantity of a
substance known to absorb those rays with great energy. Accord¬
ingly the author found that while the rays from an electric spark
which excite the phosphorescence of Canton’s phosphorus pass
freely through water and quartz, they arc stopped on adding to
the water an excessively small quantity of sulphate of quinine.

At the end of the paper the author explains what he conceives
to be the cause of the change of refrangibility, and enters into
some speculations to account for the law according to which the
refrangibility of light is always lowered in the process of internal
dispersion.

[From the Philosophical Transactions for 1852, p. 463.]

On the Change of Refrangibility of Light *

[Read May 27, 1852.]

1. The following researches originated in a consideration of
the very remarkable phenomenon discovered by Sir John Herschel
in a solution of sulphate of quinine, and described by him in
two papers printed in the Philosophical Transactions for 1845,
entitled “ On a Case of Superficial Colour presented by a Homo¬
geneous Liquid internally colourless/' and “ On the Epipolic Dis¬
persion of Light." The solution of quinine, though it appears
to be perfectly transparent and colourless, like water, when
viewed by transmitted light, exhibits nevertheless in certain
aspects, and under certain incidences of the light, a beautiful
celestial blue colour. It appears from the experiments of Sir
John Herschel that the blue colour comes only from a stratum
of fluid of small but finite thickness adjacent to the surface by

* [In choosing the title of this paper, all that was intended was, to express the
previously unsuspected origin of peculiar coloured lights which were exhibited hy
certain bodies, and which were in some cases matters of common observation, and
had been subjected to examination by physicists. The capital fact now brought
to light was that the colours in question were due to rays incident upon the body
which were of a different refrangibility from those which constituted the colour
observed. The title was meant to express a fact of observation, not any theory.
It must have been from misapprehension on this point that a view as to the nature
of the phenomenon has been attributed to me by more than one writer, a view
which I have even been said to persist in, according to which the change of re¬
frangibility takes place in the act of reflexion from the molecules of the body.
Such a view is not mine, and never was, and is directly opposed to the views
stated in the paper. The abstract, which was published some months before the
paper, contains nothing about theoretical views, but for these refers to the paper.

I can only conjecture that the view erroneously, though without hesitation, attri¬
buted to me was assumed to be mine in consequence of the title of the paper.]

268 ON THE CHANGE OF REFRANGIBTLITY OF LIGHT.

which the light enters. After passing through this stratum, the
incident light, though not sensibly enfeebled nor coloured, has
lost the power of producing the same effect, and therefore may
be considered as in some way or other qualitatively different from
the original light. The dispersion which takes place near the
surface of this liquid is called by Sir John Herschel epipolic, and
he applies the term epipolized to a beam of light which, having
been transmitted through a quiniferous solution, has been thereby
rendered incapable of further undergoing epipolic dispersion. In
one experiment, in which'sun-light was used, a feeble blue gleam
was observed to extend to nearly half an inch from the surface.
As regards the dispersed light itself, when analysed by a prism
it was found to consist of rays extending over a great range of
refrangibility: the less refrangible extremity of the spectrum was
however wanting. On being analysed by a tourmaline, it showed
no signs of polarization. A special experiment showed that the
dispersed light was perhaps incapable, at any rate not peculiarly
susceptible, of being again dispersed.

2, In a paper “ On the Decomposition and Dispersion of Light
within Solid and Fluid Bodies/' read before the Royal Society
of Edinburgh in 1846, and printed in the 16th volume of their
Transactions, as well as in the Philosophical Magazine for June
1848, Sir David Brewster notices these results of Sir John
Herschel’s, and states the conclusions, in some respects different,
at which he had arrived by operating in a different way. The
phenomenon of internal dispersion had been discovered by him
some years before, and is briefly noticed in a paper read before
the Royal Society of Edinburgh in 1833*. It is described at
length, as exhibited in the particular case of fluor-spar, in a paper
communicated to the British Association at Newcastle in 1838 f.
In Sir David Brewster’s experiments the sun’s light was con¬
densed by a lens, and so admitted into the solid or fluid to be
examined; which afforded peculiar facilities for the study of the
phenomena. On examining in this way a solution of sulphate
of quinine, it was found that light was dispersed, not merely close
to the surface, but at a long distance within the fluid : and Sir
David Brewster was led to conclude that the dispersion produced

* Edinburgh Transactions, Yol. xn. p. 542.
t Eighth Report.—Transactions of the Sections, p. 10.

ON THE CHANGE OF REFRANGIBILITY OF LIGHT.

269

by sulphate of quinine was only a particular case of the general
phenomenon of internal dispersion. On analysing the blue beam
by a rhomb of calcareous spar, it was found that a considerable
portion of it, consisting chiefly of the less refrangible rays, was
polarized in the plane of reflexion, while the more refrangible of
its rays, constituting an intensely blue beam, had a different
polarization.

3. On repeating some of Sir John HerschePs experiments, I
was immediately satisfied of the reality of the phenomenon, not¬
withstanding its mysterious nature, that is to say, that an epi-
polized beam of light is in some way or other qualitatively
different from the light originally incident on the fluid. On
making the observation in the manner of Sir David Brewster, it
seemed no less evident that the phenomenon belonged to the class
of internal dispersion * Nevertheless, the singular phenomenon
discovered by Sir John Herschel manifested itself even in this
mode of observation. If indeed the vessel containing the solution
were so placed that the image of the sun in the focus of the lens
lay a little way inside the fluid, the phenomenon was masked,
because the increase of intensity due to an increase of concen¬
tration in approaching the focus made up for the decrease of
intensity due to passing out of the blue band. But when the
vessel was moved so that the focus of the lens fell either further
inside the fluid or else outside the vessel, the narrow blue band
adjacent to the surface was seen as well as the blue beam which
shot far into the fluid. Light which has been “ epipolized ” by
transmission through a moderate thickness of the solution is
indeed capable of undergoing further dispersion, but not epipolic
dispersion, if that term, be restricted to the dispersion by which
the narrow blue band is produced. It was no doubt of great
importance to assign to the phenomenon its true place as a
member of the class of phenomena of internal dispersion. Never-

* By this, I merely mean that, to take a particular example, the exhibition of
a blue light by a solution of sulphate of quinine appeared to be a phenomenon of
the same nature as the exhibition of a red light by a solution of the green colouring
matter of leaves, although the latter does not manifest the same singular concentra¬
tion as the former in the neighbourhood of the surface by which the light enters;
and the latter had already been observed by Sir David Brewster, and the phe¬
nomenon designated as internal dispersion. I make this remark because Sir David
Brewster has applied this same term to another class of phenomena which are
totally different.

270 ON THE CHANGE OF REFRANGIBILITY OF LIGHT.

theless the mystery was by no means cleared up; rather, we were
prepared to expect something of the same sort in other instances
of internal dispersion. In fact, the mystery consisted, not in
the narrowness of the stratum from which most of the blue light
came, but in the circumstance that it was possible for light, by
passing across such a stratum, to be deprived of the power of
producing the same effect again, without, apparently, being altered
in any other respect.

4. To one who regards light as a subtle and mysterious
agent, of which the laws indeed are in a good measure known to
us, but respecting the nature of which we are utterly ignorant,
the phenomenon might seem merely to make another striking
addition to the modes of decomposition with which we were
already acquainted. But in the mind of one who regards the
theory of undulations as being for light what the theory of
universal gravitation is for the motions of the heavenly bodies,
it was calculated to excite a much more lively interest. Whatever
difficulty there might be in explaining how the effect was produced,
we ought at least to be able to say what the effect was that had
been produced; wherein, for example, epipolized light differed
from light which had not undergone that modification.

In speculating on the nature of the phenomenon, there is one
point which deserves especial attention. Although the passage
through a thickness of fluid amounting to a small fraction of an
inch is sufficient to purge the incident light from those rays which
are capable of producing epipolic dispersion, the dispersed rays
themselves traverse many inches of the fluid with perfect freedom.
It appears therefore that the rays producing dispersion are in some
way or other of a different nature from the dispersed rays produced.
Now, according to the undulatory theory, the nature of light is
defined by two things, its period of vibration, and its state of
polarization. To the former corresponds its refrangibility, and, so
far as the eye is a judge of colour, its colour*. To a change,

* It lias been maintained by some philosophers of the first eminence that light
of definite refrangibility may still be compound, and though no longer decom¬
posable by prismatic refraction might still be so by other means. I am not now
speaking of compositions and resolutions depending upon polarization. It has
even been suggested by the advocates of the undulatory theory, that possibly a
difference of properties in lights of the same refrangibility might correspond to a
difference in the law of vibration, and that lights of given refrangibility may differ

ON THE CHANGE OF B,EFR ANGIBILIT Y OF LIGHT.

271

then, either in the refrangibility or in the state of polarization we
are to look for an explanation of the phenomenon.

5. Regarding it at first as an axiom that the dispersed light
of any given refrangibility could only have arisen from light of
the same refrangibility contained in the incident beam, I was led
to look in the direction of polarization for the required change
in the nature of the light. Since a fluid has no axes, circular
polarization is the only kind which can here come into play. As
some fluids are doubly refracting, transmitting right-handed and
left-handed circularly polarized light with different velocities,
so, it might be, this fluid was doubly absorbing, absorbing say
right-handed circularly polarized light of certain refrangibilities
with great energy, and freely transmitting left-handed. The
right-handed light, absorbed, in the sense of withdrawn from the
incident beam, might have been more strictly speaking scattered,
and thereby depolarized. The common light so produced would
be equivalent to two streams, of equal intensity, one of right-
handed, and the other of left-handed circularly polarized light.
Of these the latter would be freely transmitted, while the former
would be scattered anew, and so on. Yet this hypothesis,
sufficiently improbable already, was not enough. New suppositions
were still required, to account for the circumstance that an epi-
polized beam, when subjected to prismatic analysis with a low
magnifying power, exhibited no bands of absorption in the region
to which, as regards their refrangibility, the dispersed rays prin¬
cipally belong; so that altogether this theory bore not the slightest
semblance of truth.

6. I found myself thus fairly driven to suppose that the
change of nature consisted in a change of refrangibility. From
the time of Newton it had been believed that light retains its
refrangibility through all the modifications which it may undergo.

in tint, just as musical notes of given pitch differ in quality. Were it not for the
strong conviction I felt that light of definite refrangibility is in the strict sense
of the word homogeneous, I should probably have been led to look in this direction
for an explanation of the remarkable phenomena presented by a solution of
sulphate of quinine. It would lead me too far from the subject of the present
paper to explain the grounds of this conviction. I will only observe that I have
not overlooked the remarkable effect of absorbing media in causing apparent
changes of colour in a pure spectrum; hut this I believe to be a subjective
phenomenon, depending upon contrast.

272 ON THE CHANGE OF REFRANGIBILITV OF LIGHT.

Nevertheless it seemed to me less improbable that the refrangi-
bility should have changed, than that the undulatory theory
should have been found at fault. And when I reflected on the
extreme simplicity of the whole explanation if only this one
supposition be admitted, I could not help feeling a strong expect¬
ation that it would turn out to be true. In fact, we have only
to suppose that the invisible rays beyond the extreme violet give
rise by internal dispersion to others which fall within the limits
of refrangibility between which the retina of the human eye
is affected, and the explanation is obvious. The narrowness of
the blue band observed by Sir John Herschel would merely
indicate that the fluid, though highly transparent with regard
to the visible rays, was nearly opaque with regard to the invisible.
According to the law of continuity, the passage from almost
perfect transparency to a high degree of opacity would not take
place abruptly; and thus rays of intermediate refrangibilities
might produce the blue gleam noticed by Sir John Herschel, or
the blue cylinder, or rather cone, observed by Sir David Brewster.
We should thus, too, have an immediate explanation of a remark¬
able circumstance connected with the blue band, namely that
it can hardly be seen by strong candle-light, though readily seen
by even weak daylight. For candle-light, as is well known, is
deficient in the chemical rays situated beyond the extreme violet.

7. My first experiments were made with coloured glasses.
A test tube was about half filled with a solution consisting of
disulphate of quinine dissolved in 200 times its weight of water
acidulated with sulphuric acid. The tube, having been first
covered with black paper, with the exception of a hole by which
the light might enter, wtis placed in a vertical position in front of
a window, the hole being turned towards the light. On looking
down from above, in a direction nearly parallel to the surface
of the glass, a blue arc was well seen, extending only a very
short distance into the fluid, and situated immediately behind
the hole. As this arc, though extremely distinct, was not of
course what could be called brilliant, I did not at first venture,
for the experiment I had in view, to use any but pale glasses.
Having no direct means of determining which were opaque with
regard to the invisible rays situated beyond the extreme violet,
I sought among a collection of orange, yellow, and brown glasses,
which, from transmitting mainly the less refrangible rays, seemed

ON THE CHANGE OF REFRANGIBILITY OF LIGHT.

273

the most likely to absorb the chemical rays. I presently found a
pale smoke-coloured glass, which, when placed immediately in
front of the hole, prevented the formation of the blue arc, although
when placed immediately in front of the eye it transmitted a
large proportion of the light of which the arc consisted. The
colour of the arc was of course modified, and rendered more
nearly white.

On trying other pale glasses, I found one of a puce colour,
which, when placed in front of the hole, allowed the arc to be
formed, though it absorbed it when placed in front of the eye.
A yellow, and likewise a yellowish green glass allowed the arc
to be seen in both positions ; but its colour was decidedly different
according as the glass was placed in front of the hole or in front of
the eye. The breadth, too, of the arc was differently affected by
different coloured glasses placed in front of the hole, some causing
the light to be more, and others less concentrated towards the
surface of the test tube than when the incident light was un¬
impeded.

8. The sun’s light was next reflected horizontally into a
darkened room, and allowed to pass through a hole in a vertical
board which was placed in the window. The hole contained a
lens of rather short focus. On placing a test tube containing the
solution, in a vertical position, in front of the lens, at such a
distance that the focus lay some way inside the fluid, the narrow
blue band described by Sir John Herschel and the blue beam
mentioned by Sir David Brewster were seen independently of each
other. On trying different coloured glasses, which were placed,
first in front of the fluid, and then in front of the eye, it was found
that the blue beam, as had previously proved to be the case with
the narrow band, was for the most part differently affected according
as the glass was placed so as to intercept the incident or the dis¬
persed light. Moreover, the long blue beam and the narrow band
did not behave in the same manner under the action of the same
coloured glass.

9. To my own mind these experiments were conclusive as to
the fact of a change of refrangibility. Admitting that the effect
of a coloured glass is simply to stop a certain fraction of the
incident light, that fraction being a function of the refrangibility,
it is plain that the results can be explained in no other way. It

18

s. in.

274

ON THE CHANGE OF REFRANGIBILITY OF LIGHT.

must be confessed however that these results are merely an exten¬
sion of that which precisely constitutes the peculiarity of the
phenomenon. For, take the case of the narrow blue band formed
by ordinary daylight. Imagine a glass vessel with parallel sides
to be filled with a portion of the solution, and placed so as to
intercept, first the incident, and then the dispersed light. In the
first position the light incident on the fluid under examination
would be “ epipolized” by transmission through the fluid con¬
tained in the vessel, and therefore the blue band would be cut off,
whereas when the vessel was held in front of the eye the blue band
would be freely transmitted. Hence the effects of the coloured
glasses are analogous to, but less striking than, the effect of a
stratum of the solution of sulphate of quinine in the imaginary
experiment above described. There is to be sure one important
difference in the two cases, namely, that in the case of the stratum
of fluid the epipolic dispersion which is prevented in the fluid
under examination is produced near the first surface of the stratum,
whereas no such dispersion is produced, or at any rate necessarily
produced, in the coloured glasses. Whatever the reader may
think of the results obtained with coloured glasses, the next ex¬
periment it is presumed will be deemed conclusive.

10. The board in the window containing the lens having been
replaced by a pair of boards adapted to form a vertical slit, the
sun’s light was reflected horizontally through the slit, and trans¬
mitted through three Munich prisms placed one after the other
close to it. A tolerably pure spectrum was thus formed at the dis¬
tance of some feet from the slit. A test tube containing the
solution was then placed vertically a little beyond the extreme red
of the spectrum, and afterwards gradually moved horizontally
through the colours. Throughout nearly the whole of the visible
spectrum the light passed through the fluid as it would have done
through so much water; but on arriving nearly at the violet ex¬
tremity a ghost-like gleam of pale blue light shot right across the
tube. On continuing to move the tube, the blue light at first
increased in intensity and afterwards gradually died away. It did
not however cease to appear until the tube had been moved far
beyond the violet extremity of the spectrum visible on a screen.
Before disappearing, the blue light was observed to be confined to
an excessively thin stratum of fluid adjacent to the surface by
which the light entered, whereas when it first appeared, namely

ON THE CHANGE OF REFRANGIBILITY OF LIGHT.

275

when the tube was placed a little short of the extreme violet, the
blue light had extended completely across it. It was certainly
a curious sight to see the tube instantaneously lighted up when
plunged into the invisible rays: it was literally darkness visible .
Altogether the phenomenon had something of an unearthly
appearance.

11. Since the fluid is so intensely opaque with regard to rays
of extreme refrangibility, it might be expected, that, though it
appears transparent and colourless when examined merely by view¬
ing a white object through it, it would yet exhibit a very sensible
absorbing action with regard to the extreme violet rays when
subjected to prismatic analysis. To try whether such were really
the case, I reflected the sun’s light horizontally through a slit,
at which was placed a test tube filled with the liquid, and analysed
the line of light by a prism, the eye being defended by a deep
blue glass. I was barely able to make out the fixed line H,
that is, the less refrangible band of the pair, although in similar
circumstances I can generally see about as far beyond the more
refrangible band as it is beyond H. However, to make the
result more decisive by using a greater thickness, as well as to
render the observation strictly differential, I placed a tumbler filled
with water behind the slit, the blue glass before it, and then
viewed the slit through the prism. I saw as far as usual into the
violet. The water was then poured out and replaced by the
solution of sulphate of quinine, which, when viewed by transmitted
light, appeared as transparent as the water which it had replaced.
When the tumbler was now placed behind the slit, the blue beam
of dispersed light was observed to extend quite across it, a dis¬
tance of about three inches, and would evidently have gone much
further. On viewing the slit through the prism, the spectrum
was found to be cut off about half-way between the fixed lines G
and II. The termination was pretty definite, which indicates that,
at least for that part of the spectrum, the absorbing energy of the
fluid rapidly increased with the refrangibility of the light; there
was, however, an evident diminution of intensity produced by the
fluid, extending from the termination of the spectrum to near G.

12. There could no longer be any doubt, either as to the
fact of a change of refrangibility, or as to the explanation thereby
of the remarkable phenomenon exhibited by sulphate of quinine.

18—2

276

ON THE CHANGE OF REFRANGIBILITY OF LIGHT.

Epipolized light is merely light which has been purged of the
invisible, or at most feebly illuminating rays more refrangible
than the violet; and the term itself, which in fact was only
adopted provisionally by Sir John Herschel, and which has now
served its purpose, may henceforth be discarded, especially as it
is calculated to convey a false impression respecting the cause of
the phenomenon. It remained to examine other instances of
internal dispersion, of which, according to Sir David Brewster’s
observations, the dispersion produced by sulphate of quinine is
only a particular case; to endeavour to make out the laws accord¬
ing to which a change of refrangibility takes place; and, if
possible, to account for these laws on mechanical principles.

13. In giving an account of my further experiments, I think
it best to describe in detail the phenomena observed in some of
the more remarkable instances of internal dispersion before at¬
tempting to draw any general conclusions. It will save repetition
to explain in the first instance the methods of observation em¬
ployed, which on the whole may very fairly be divided into four,
though occasionally it was convenient to employ intermediate
methods, or a combination of two of them. Of course I fre¬
quently availed myself of Sir David Brewster’s method of obser¬
vation, in which the effect of the incident light is studied as a
whole; but the methods here referred to relate to an investigation
of the separate offices of the portions of light of different degrees
of refrangibility which are found in the incident beam. As my
researches proceeded, new methods of observation suggested them¬
selves, but these will be described in their place.

Methods of Observation employed.

First Method. —The sun’s light was reflected horizontally
through a small lens, which was fixed in a hole in a vertical
board. The cone of emergent rays was allowed to enter the solid
or fluid examined. A coloured glass or other absorbing medium
was then placed, first so as to intercept the incident rays, and
then between the substance examined and the eye. For short¬
ness’ sake these positions will be designated as the first and the
second. Sometimes a coloured glass was allowed to remain in
front of the hole, and a second glass was added, first in front of
the hole and then in front of the eye.

ON THE CHANGE OF REFRANGIBILITY OF LIGHT.

277

Second Method.— The sun’s light, reflected as before, was
transmitted through a series of three or four Munich prisms
placed one immediately after the other, and each nearly in the
position of minimum deviation. It was then transmitted through
a small lens in a board close to the last prism, and so allowed to
enter the body to be examined, which was generally placed so
that the first surface coincided, or nearly so, with the focus of the
lens. The diameter of the lens was much smaller than the
breadth or height of the prisms, so that the lens was completely
filled with white light, the component parts of which however
entered in different directions. Regarding the image of the sun
in the focus of the small lens as a point, we may conceive the
light incident on the body under examination as consisting of a
series of cones, corresponding to different refrangibilities, the axes
of which lay in a horizontal plane and intersected in the centre
of the lens, the vertices being arranged in a horizontal line near
the surface of the body examined.

Third Method.— The sun’s light was reflected horizontally
through a vertical slit, and received on the prisms, which were
arranged as before, but placed at the distance of several feet from
the slit. A large lens of rather long focus was placed immediately
after the last prism, with its plane perpendicular, or nearly so,
to the beam of light which had passed through the prisms, and
with its centre about the middle of this beam. The body
examined was placed at the distance of the image of the slit, or
nearly so.

Fourth Method.— Everything being arranged as in the third
method, a board with a small lens of short focus was placed at
the distance of the image of the slit, or between that and the
image of the sun, which was a little nearer to the prisms, in¬
asmuch as the focal length of the large lens commonly employed,
though much smaller, was not incomparably smaller than the
distance of the lens from the slit. A second slit was generally
added immediately in front of the small lens. The body examined
was placed at the focus of the small lens. The dispersed light
was viewed from above, and analysed by a prism, being refracted
sideways.

The object of these several arrangements will appear in the
course of the paper. The prisms employed consisted, three of

278 ON THE CHANGE OF 11EFR ANGIBILITY OF LIGHT.

them of flint glass and one of crown. The refracting angles of
the former were about 43°, 33°, and 24°, and that of the latter
about 45°. The refracting faces of the smallest of the prisms
(the flint of 43°) were T35 inch high and 1*60 long. The small
lens used was one or other of a pair of which the apertures were
0*34 inch and 0*22 inch, and the focal lengths 0*75 inch and
0*50 inch. The focal length of the large lens generally used was
about twelve inches. Once or twice a lens was tried which had
a focal length about three times as great, but the light proved
too faint for most purposes. In the third method it was some¬
times convenient to employ a lens of only 6|~ inches focal length,
but the 12-inch lens was employed in the fourth method, except
on a few occasions, when the lens of 36 inches focal length was
used. With the 12-inch lens the length of the spectrum from the
fixed line B to H was usually about an inch and a quarter.

It will be convenient for the purposes of this paper to employ
certain terms in a particular sense, but as some of these terms
relate to phenomena which have not yet been described, it will
be well previously to relate in detail what was observed in one
remarkable instance of internal dispersion.

Solution of Sulphate of Quinine .

14. The effects of some pale coloured glasses in the case
of this fluid have already been mentioned. But there is one glass
of which the effect is still more striking. It is well known that
a deep cobalt blue glass is highly transparent with regard to the
chemical rays. Accordingly I found that a blue glass, so deep
that only the brighter objects in a room could be seen through
it, produced but very little effect when placed so as to intercept
the light incident on the fluid. When placed immediately in
front of the eye, at first everything disappeared except the light
reflected from the convexities of the glass tube; but when the
eye became a little accustomed to the darkness it was possible
to make out the existence of the band. The contrast between
the effects of this glass and of the pale brown glass already
mentioned was most striking.

15. When the fluid was examined by the second method,
the dispersed light was found to consist of two beams, separated

ON THE CHANGE OF REFRANGIBILITY OF LIGHT. 279

from each other at their entrance into the fluid, that is, at the
vertical surface of separation of the fluid and the containing
vessel, and afterwards still further separated by divergence. Of
course each beam must have been made up of a series of cones
having their axes diverging from the centre of the lens, and their
vertices situated at its focus. The first beam, or that which was
produced by light of less refrangibility, consisted of the brighter
colours of the spectrum in their natural order. It had a dis¬
continuous, sparkling appearance, and was plainly due merely to
motes which were suspended in the fluid. On being viewed from
above through a Nicol’s prism, it was found to consist chiefly
of light polarized in the plane of reflexion. Taken as a whole,
it served as a fiducial line to which to refer the position of the
second beam, and thereby judge of the refrangibility of the rays
by which it was produced.

This second beam was a good deal the brighter of the two.
Its colour was a beautiful sky-blue, which was nearly the same
throughout, but just about its first border, that is, where it arose
from the least refrangible of those rays which were capable of
producing it, the colour was less pure. It had a perfectly con¬
tinuous appearance. When viewed from above through a doubly
refracting achromatic prism of quartz, which allowed a direct
comparison of the two images, it offered no traces of polarization.
It was produced by light polarized in a vertical or horizontal
plane as well as by common light, and in that case, as well as
in the former, manifested no traces of polarization *.

The short distance that the more refrangible rays were able
to penetrate into the fluid might readily be perceived in this
experiment, but the second method of observation was not adapted
to bring out this part of the phenomenon.

1G. On examining the fluid by the third method, the result
was very striking, although of course only what might have been

:: These two results, namely, that the blue beam which constitutes the greater
part of the light dispersed by a solution of sulphate of quinine is unpolarized,
or according to his expression possesses a quaquaversns polarization, and that
that still remains the case when the incident light is polarized, have been already
announced by Sir David Brewster, who appears to have been led to attend to the
polarization of the light from Sir John Herschel’s observation, that the blue light
arising from epipolic dispersion in a solution of sulphate of quinine was un-
polarized. It seemed important however to repeat the observation on the blue
beam obtained in a state of isolation.

280

ON THE CHANGE OF KFFRAX<UBILITY OF LIGHT.

anticipated. The principal fixed lines of the violet, and of the
chemical parts of the spectrum beyond, were seen with beautiful
distinctness as dark planes interrupting an otherwise perfectly
continuous mass of blue light. To see any particular fixed line
with most distinctness, it was of course necessary to hold the
eye in the corresponding plane, when the dark plane was fore¬
shortened into a dark line. From the red end of the spectrum,
as far as the line (r, or thereabouts, the light, passed freely
through the fluid, or at least was only reflected here and there
from motes held in mechanical suspension. About G the dis¬
persion just commenced to be sensible, and there were traces of
that line seen as a dark plane interrupting a mass of continuous
but excessively faint light. For some distance further on the
dispersed light remained so faint that it might haves been passed
over if not specially looked for. If was about half-way between
G and II, or a little before, that it. first became so strong as to
arrest attention, and a little further on it became very con¬
spicuous, the tint meanwhile changing to a pale sky-blue. The
light was very copious about tin* two broad bands of the group II
and for some distance from II towards (}. Some of the fixed
lines less refrangible than II wen; very plain, and beyond II
a good number were visible, which will presently lx; further
described. The whole system of fixed lines thus visible as
interruptions in the dispersed light had a, resolvable appearance;
but with a very narrow slit, and a lens of long focus at the; prisms
the light would have been too faint for convenient, observation.

The dispersed light about (7, and for some distance further
on, was so very faint that I might have; overlooked it had it
not arrested my attention when observing by the fourth method;
indeed, I have sometimes spc‘ciallv looked for it in the third
arrangement without having been able to see it. Practically
speaking, the dispersion might be said to commence, about half¬
way between G and II.

17. On refracting the whole system sideways through a prism
of moderate angle held in front of the eye, the fixed lines became
confused, and the finer ones disappeared. The edges of the broad
bands H were tinged with prismatic colours, like the edges of
two slips of black velvet placed on a sheet of pale blue paper,
and viewed through a prism. This experiment exhibits the

ON THE CHANGE OF RE FRAN GIBI LIT Y OF LIGHT.

281

compound character of the dispersed light, notwithstanding the
perfect homogeneity of the incident light.

18. The third method of observation is well adapted to bring
into view the variation in the absorbing energy of the medium
corresponding to a variation in the refrangibility of the incident
rays. When the eye is placed vertically over the vessel con¬
taining the solution, so that the dark planes corresponding to the
fixed lines of the spectrum are projected into dark lines, of which
the length is not exaggerated by obliquity, the boundary of the
dispersed light is projected into a curve, which serves to represent
to the eye the relation between the absorbing power of the
medium and the refrangibility of the incident light. This curve
is not exactly that which Sir John Herschel has treated of in
the theory of absorption, and considered as the type of the
absorbing medium to which it is applied, but nevertheless it
serves much the same purpose. It is true, that, independently
of any change in the absorbing energy of the medium, an in¬
creasing faintness in the dispersed light would produce to a
certain extent an approximation of the curve to its axis; but
practically, in the case of sulphate of quinine, as well as in a
great many others, the appearance is such as to leave no doubt
as to the existence of a most intense absorbing energy on the
part of the medium with respect to rays of very high re-
frangibilities*.

In the case of a solution of sulphate of quinine of the
strength of one part of the disulphate to 200 parts of acidulated
water, it has been already stated that a portion of the rays
which are capable of producing dispersed light passed across a
thickness of 3 inches. On forming a pure spectrum, the fixed
line H was traced about an inch into the fluid. On passing
from H towards 6r, the distance that the incident rays penetrated
into the fluid increased with great rapidity, while on passing
in the contrary direction it diminished no less rapidly, so that
from a point situated at no great distance beyond H to where
the light ceased, the dispersion was confined to the immediate

* I should here remark, that, after the researches described in this paper had
far advanced, I met accidentally with a passage in the Competes Rendus , tom. xvii.
p. 883, in which M. Ed. Becquerel mentions a solution of acid sulphate of quinine
as a medium eminently remarkable for its absorbing power with respect to the
rays more refrangible than H.

282

ON THE CHANGE OF IIEFHANGIBIJLITY OF LIGHT.

neighbourhood of the surface. When the solution was diluted
so as to be only one-tenth of the former strength, a conspicuous
fixed line, or rather band of sensible breadth, situated in the
first group of fixed lines beyond H, was observed to penetrate
about an inch into the fluid. On passing onwards from the band
above-mentioned in the direction of the more refrangible rays,
the distance that the incident rays penetrated into the fluid
rapidly decreased, and thus the rapid increase in the absorbing
energy of the fluid was brought into view in a part of the
spectrum in which, with the stronger solution, it could not be
so conveniently made out, inasmuch as the posterior surface of
the space from which the dispersed light came almost confounded
itself with the anterior surface of the fluid.

The high degree of opacity with regard to rays of great
refrangibility which the addition of so small a proportion of
sulphate of quinine is sufficient to produce in water is certainly
very remarkable; nevertheless it is only what 1 have constantly
observed while following out these researches.

19. In observing by the fourth method, the part of the
spectrum to which the incident light belonged was determined
sometimes by the colour, sometimes by moans of the fixed lines
of the spectrum. It almost always happened that there were
motes enough suspended in the fluid to cause a portion of the
dispersed beam to consist merely of light which had undergone
ordinary reflexion. When the whole dispersed beam was analysed
by a prism, the beam which consisted of light reflected from
motes was separated from the rest; it was in general easily
recognised by its sparkling appearance, but at any rate was
known by its consisting almost wholly of light polarized in the
plane of incidence, whereas the truly dispersed light was un¬
polarized. It consisted of course of light of definite refrangibility,
the same as that of the incident light, and thus served as a
fiducial line to which to refer by estimation the refrangibilities
of the component parts of the dispersed light. Of course this
part of the observation was possible only when the incident rays
belonged to the visible part of the spectrum.

On moving the lens horizontally through the colours of the
spectrum, in a direction from the red to the violet, it was found
that the dispersion was first perceptible in the blue. When the

ON THE CHANGE OF REFRANGIBILITY OF LIGHT.

283

dispersed light was separated by a prism from the light reflected
from motes, it was found to consist of an exceedingly small
quantity of red; further on some yellow began to enter into its
composition ; further still, perhaps about the junction of the
blue and indigo, the dispersed beam began to grow brighter, and
was found on analysis to contain some green in addition to the
former colours. In the indigo it got still brighter, and when
viewed as a whole was somewhat greenish. Further still it
became something of a pale slaty blue, and was found on analysis
to contain some indigo, or at least highly refrangible blue. On
proceeding further the dispersed light became first of a deeper
blue and then, a little short of the fixed line H, whiter. At a
considerable distance beyond H the dispersed light was if any¬
thing a shade more nearly white.

By this method of observation the dispersion can be detected
earlier in the spectrum than by the third method, and moreover
the change in the colour of the dispersed light is much more
easily perceived; indeed the most striking part of this change
takes place while the dispersed light is so very faint that it can
hardly be seen in observing by the third method; moreover, even
in the bright part of the dispersed beam, it is not at all easy by
the latter method to make out the change of tint corresponding
to a change in the refrangibility of the incident rays, because the
tint changes so gradually and so slightly that the eye glides from
one part of the dispersed beam to another without noticing any
change.

20. It has been already mentioned that the blue beam of
dispersed light seen in a solution of sulphate of quinine was pro¬
duced whether the incident light was polarized in or perpendicu¬
larly to the plane of reflexion, or more properly plane of dispersion,
that is, the plane containing the incident ray and that dispersed
ray which enters the eye. A question naturally presents itself,
whether the intensity of the dispersed light is strictly the same
in the two cases. By combining a lens of rather short focus and
a doubly refracting prism with the four prisms, I satisfied myself
that the difference of intensity, if there were any, was not great,
but the experiment presented some practical difficulties. How¬
ever, the result of the following experiment appeared to be as
decisive as a negative result could well be.

ON THE GHAXGF OF REFRAINUBIIJTY OF LIGHT.

2<S4

The arrangement being the same as in the third method, but
the lens in front of the prisms having a focal length of only
6*5 inches, the incident light was polarized in a vertical plane
previously to passing through the slit, by transmission through a
pile of plates. The two beams of light were seen as usual in the
fluid, namely, the blue beam due to internal dispersion, and the
fainter coloured beam due to motes. The former of these, which
was quite separate from the latter, exhibited the principal fixed
lines belonging to the highly refrangible part of the spectrum. A
plate of selenite was then interposed immediately in front of the
vessel, so as to modify the polarization of the light entering the
fluid. This plate was obtained by an irregular natural cleavage,
and was cemented with Canada balsam between two discs of
glass. When examined by polarized light it exhibited a succession
of beautiful and varied tints, according to the various thicknesses
of the different parts. Now when the plate was moved about in
front of the vessel, without altering its perpendicularity to the
incident light, different portions of the beam due to motes were
observed to disappear and reappear, or at least to become faint
and then bright again, so that a person ignorant of the cause, and
not looking at the disc, might have supposed that the observer
had been holding in front of the vessel a piece of dirty glass,
having the dirt laid on in patches; but in whatever manner the
disc was moved in its own plane without rotation, or turned round
an axis perpendicular to its plane, not the slightest perceptible
change was produced in any part of the blue beam.

Explanation of Terms.

21. In all the experiments described in this paper in which a
spectrum was formed for the sake of examining the separate action
of portions of light of different refrangibililies, the length of the
spectrum was horizontal, so that the fixed lines were vertical.
Nevertheless it will be convenient, for the sake of shortness, to
use the prepositions above and below to signify respectively on th,e
more refrangible side of and on the less refrangible side of.

The principal fixed lines of the visible spectrum will be denoted
by letters in accordance with Fraunhofer’s admirable map. These
lines are now too well known to need description.

ON THE CHANGE OF REFRANGIBILITY OF LIGHT.

285

The only map of the fixed lines of the chemical spectrum
which I had for a good while after these researches were com¬
menced is Professor Draper s, which will he found in the twenty-
second volume of the Philosophical Magazine (1843). Of course
this map cannot he compared for accuracy of detail with Fraun¬
hofer’s map of the visible spectrum, nor does it profess to give
more than some of the most conspicuous lines selected from among
a great multitude. The suppression of so many lines, without
any representation by shading of their general effect, renders it
difficult to identify those which are laid down, at least if I may
judge from my own observations; besides, Professor Draper’s
spectrum was so much purer than the one with which I found it
most convenient to work, that the two are not comparable with
each other.

22. I have made a sketch of the fixed lines from H to the end,
which accompanies this paper (see Plate). The fixed lines of the
visible spectrum are so well known that I thought it unnecessary
to begin before H. A solution of sulphate of quinine is a very good
medium for showing the lines, but a yellow glass, which will be
mentioned presently, is quite as good, or rather better. The map
represents the spectrum as seen with the lens of 12 inches focal
length in front of the prisms. The breadth of the slit was not
always quite the same: it maybe estimated at about the J^th of an
inch. The map contains 32 fixed lines or bands more refrangible
than II, which is the utmost that I have been able on different
occasions to see with this lens, though with a lens of longer focus
and a narrower slit the number of fixed lines which might be
counted was, as might be expected, a good deal larger. As I have
not yet identified these lines, except in certain cases, with those
which had previously been represented by means of photographic
impressions, I have thought it advisable not to attempt an identi¬
fication, but to attach letters to the more conspicuous lines in my
map without reference to former maps. As the capitals Z, M, N,
(), P have already been appropriated to designate certain fixed
lines, I have made use of the small letters ?, m, n, o, p, to prevent
confusion.

In drawing the map, I have endeavoured to preserve the
character of the lines with respect to blackness or faintness, sharp¬
ness or diffuseness. The distances were not laid down by measure-

286 ON THE CHANGE OF REFRANGIBILITY OF LIGHT.

ment, except here and there, and they are not, I fear, quite so
accurate as might be desired; still, I feel assured that no one
viewing the actual object would feel any difficulty in identifying
the lines with those in my map, provided the circumstances under
which his spectrum was formed at all approached to those under
which mine was seen when the arrangement as to focal length of
the lens, &c. was that most convenient for general purposes.

The more conspicuous lines in the part of the spectrum repre¬
sented in the map may conveniently be arranged in five groups,
which I will call the groups H, l , m, n t p. The group H consists
chiefly of the well-known pair of bands of which the first contains
Fraunhofer’s line H ; the second band I have marked k, in accord¬
ance with Professor Draper’s map. The most conspicuous object
in the next group consists of a broad dark band, l. This band is
between once and twice as broad as H, and is darker in the less
refrangible half than in the other. With a lens of 3 feet focal
length and a narrow slit it was resolved into lines, which is
probably the reason why it is altogether omitted in Professor
Draper’s map, while the first three lines of the group (if I do not
mistake as to the identification) are represented, forming his
group L. Under the circumstances to which the accompanying
map corresponds, the band l appears as a very striking object,
perhaps, with the exception of the bands H , k, the most con¬
spicuous in the whole spectrum. With a still lower power it
appears as a very black and conspicuous line. A double line
beyond l completes the group l, after which comes another remark¬
able group m, consisting of five lines or bands. Of these the first
is rather shady, though sharply cut off on its more refrangible
side, but the others, and especially I think the second and third,
are particularly dark and well-defined. I have marked the middle
line m, not because it is more conspicuous than its neighbours,
but on account of its central situation. After a very faint group,
consisting apparently of four lines, comes another very conspicuous
group n 9 consisting of two pairs of dark bands followed by another
pair of bands which are broad and very dark. The first of these
is a good deal broader than the second, but is not so broad as the
band H ; the second is followed by a fine line. This is as far as it
is easy to see; but when the sunshine is clear, and the arrange¬
ments are made with a little care, a group of six lines is seen
much further on. Of these, the first two are only moderately

ON THE CHANGE OF REFRANGIBILITY OF LIGHT.

287

dark, and the first is rather diffuse; they stand off a little from
the others, and are a little closer together than the other four.
Of the latter, the first, marked o, is very strong, considering the
faintness of the light which it interrupts; the second and third
are faint, and difficult to see; the fourth, marked p, is black like
the first, and a good deal broader. The line p was situated, by
measurement, as far beyond H as H beyond b. Once or twice in
the height of summer, and under the most favourable circum¬
stances, I have observed two broad dusky bands still further on.
The first of these had the appearance of being resolvable into two.
The excessively faint light seen beyond the second seemed to end
rather abruptly at the distance represented by the border of the
accompanying plate, as if there were there the edge of another
dark band beyond which nothing could.be seen. In order to see
the dusky bands last mentioned, and even to see the group p to
most advantage, it was necessary to allow the central part of the
beam incident on the prisms to pass through them close to their
edges, so that evidently a great deal of light was lost by passing
by the prisms altogether. This circumstance, combined with
others which I have observed, convinces me that the great obstacle
to seeing the fixed lines in this part of the spectrum consists
in the opacity of glass. Were glass as transparent with respect to
the invisible rays of very high refrangibility as it is with respect
to the rays belonging to the visible spectrum, I know not how
much further I might have been able to see.

I have endeavoured to identify the fixed lines in my map with
the fixed lines represented in M. Silbermann’s map of the chemical
spectrum, with a copy of which my friend Professor Thomson has
kindly furnished me. I am still uncertain respecting the identifi¬
cation. M. Silbermann’s map is so very much more detailed than
my own, and must have been made with so much purer a spec¬
trum, that the two systems of lines are not directly comparable.

23. From the difficulty of identification some persons might
be disposed to imagine that the chemical rays, and those which
produced the blue light in a solution of quinine, were of a different
nature, and had each a system of fixed lines of its own. For my
own part, 1 was too well acquainted with the Protean character of
fixed lines to regard the difficulty of identification as any valid
argument in support of such a view. And that this difficulty

288

ON THE CHANGE OF REFRANG1BILITY OF LIGHT.

arose from nothing more than the different degrees of purity of
the spectra is now put past dispute, for my friend Mr Kingsley of
Sidney Sussex College, to whom I recently showed some of the
experiments mentioned in this paper, has kindly taken for me
some photographs of spectra having nearly the same degree of
extent and purity as those with which I worked, and these show
the fixed lines just as they appeared in a solution of sulphate of
quinine and in other media*.

24. The position of a point in the spectrum which does not
coincide with one of the principal fixed lines, will be denoted by
referring it to two of those lines, in a manner which will be most
easily explained by an example. Thus \GH> G^H, GH^ will be
used to denote respectively a point situated at a distance below G
equal to half the interval from G to II , a point midway between
G and H ) and a point situated at the same distance above H.
In using this notation, the letters denoting fixed lines will be
written in the order of refrangibility, and the fraction expressing
the part of the interval between these lines, which must be con¬
ceived to be measured off in order to reach the point whose
position it is required to express, will be written before, between,
or after the letters, according as the measurement is to be taken
from the first line in the negative direction, from the first line in
the positive direction, or from the second line in the positive
direction, the positive direction being that of increasing refrangi¬
bility.

25. From the experiments already described, it appears that
the beam of dispersed light which was observed in the experi¬
ments of Sir David Brewster consisted of two very distinct por¬
tions, one arising merely from light reflected from motes, and the
other having a far more remarkable origin. It will be convenient
to have names for these two kinds of dispersion, and I shall
accordingly call them respectively false internal dispersion and
true internal dispersion , or simply false dispersion and true disper¬
sion when the context sufficiently shows that internal dispersion
is spoken of. When dispersion is mentioned without qualification,
it is to be understood of true dispersion. Now that it appears
that the mere reflexion of light from solid particles held in
mechanical suspension has nothing to do with that remarkable

* See note A at the end.

ON THE CHANGE OF REFRANGIBILITY OF LIGHT.

289

kind of internal dispersion which is characterized by the “ quaqua -
versus polarization/’ the phenomenon of false dispersion ceases to
he of much interest in an optical point of view; while on the
other hand the phenomenon of true dispersion, which had always
been very remarkable, is now calculated to excite a great addi¬
tional interest. It will be convenient to mention here the princi¬
pal characters by which true and false dispersion may be distin¬
guished, although it will he anticipating in some measure the
results of observations yet to he described.

2(j. In true dispersion the dispersed light has a perfectly con¬
tinuous appearance. In false dispersion, on the other hand, it has
generally more or less of a sparkling appearance, and on close
inspection is either wholly resolved into bright specks, or so far
resolved as to leave on the mind the impression that if the resolu¬
tion be not complete it is only for want of a sufficient magnifying
power.

In true dispersion the dispersed light is perfectly unpolarized.
In false dispersion, on the contrary, at a proper inclination the
light is almost perfectly polarized in the plane of reflexion.

In false dispersion, which is merely a phenomenon of reflexion,
the dispersed light has of course the same refrangibility as the in¬
cident light. In true dispersion heterogeneous dispersed light
arises from a homogeneous beam incident on the body by which
the dispersion is produced.

27. In those bodies, whether solid or liquid, which possess in
a high degree the power of internal dispersion, the colour thence
arising may be seen by exposing the body to ordinary daylight,
looking at it in such a direction that the regularly reflected light
does not enter the eye, and excluding transmitted light by placing
a piece of black cloth or velvet behind, or by some similar con¬
trivance. It has been usual to speak of the colour so exhibited as
displayed by reflexion. As however the cause now appears to be
so very different from ordinary reflexion, it seems objectionable to
continue to use that term without qualification, and I shall accord¬
ingly speak of the phenomenon as dispersive reflexion * Thus

* I confess I do not like this term. I am almost inclined to coin a word, and
call the appearance fluorescence , from fluor-spar, as the analogous term opalescence
is derived from the name of a mineral.

S. III.

19

290 ON THE CHANGE OF KEFRANGIBILITY OF LIGHT.

dispersive reflexion is nothing more than internal dispersion con¬
sidered as viewed in a particular way.

28. The tint exhibited by dispersive reflexion is modified in
a peculiar manner by the absorbing power of the medium. In
the first place, the light which enters the eye in a given direction
is made up of portions which have been dispersed by particles
situated at different distances from the surface at which the light
emerges. The word particle is here used as synonymous, not with
molecule , but with differential element If we consider any parti¬
cular particle, the light which it sends into the eye has had to
traverse the medium, first in reaching the particle, and then in
proceeding towards the eye. On account of the change of re-
frangibility which takes place in dispersion, the effect of the
absorption of the medium is different for the two portions of the
whole path within the medium, so that this effect may be regarded
as a function of two independent variables, namely, the lengths of
the path before and after dispersion; whereas, had the light been
merely reflected from coloured particles held in suspension, the
effect of absorption would have been a function of only one in¬
dependent variable, namely, the length of the entire path within
the medium.

29. When false dispersion abounds in a fluid, it may be
detected at once by the eye, without having recourse to any of
the characters already mentioned whereby it may be distinguished
from true dispersion. When a fluid is free from false dispersion
it appears perfectly clear, when viewed by transmitted light,
although it may be highly coloured, and may even possess to
such an extent the property of exhibiting true internal dispersion
as to display, when properly viewed, a copious dispersive reflexion.
On the contrary, when false dispersion abounds, the fluid, if not
plainly muddy, has at least a sort of opalescent appearance when
viewed by transmitted light, which, after a little experience, the
eye in most cases readily recognizes. In viewing the phenomenon
of dispersive reflexion, as exhibited in a fluid, it might be supposed
that the fluid was water, or else some clear though coloured liquid,
holding in suspension a water colour in a state of extreme sub¬
division. But on holding the fluid before the eye, so as to view it
by transmitted light, or rather view a bright well-defined object
through it, the illusion is instantly dispelled. The reason of this

ON THE CHANGE OF REFRANGIBILITY OF LIGHT.

291

difference appears to admit of easy explanation, and will be noticed
further on.

30. Light will be spoken of in this paper as active when it is
considered in its capacity of producing other light by internal
dispersion. A medium will be said to be sensitive when it is
capable of exhibiting dispersed light under the influence of light
(visible or invisible) incident upon it. In the contrary case it
will be called insensible.

I shall now return to the description of the appearances ex¬
hibited by some of the media most remarkable for their sensibility.

Decoction of the Bark of the Horse-Chestnut (JEsculus hippo-

castanum).

31. In Sir John Herschel’s second paper it is stated that
esculine possesses in perfection the peculiar properties which had
been found to belong to quinine. Having tried without success to
procure the former alkaloid* I was content to let this substance
pass, till I found how admirably a mere decoction or infusion of
the bark of the tree answered for all purposes of observation.

This medium is even more sensitive than a solution of sulphate
of quinine, and disperses like it a blue light. The description of
the mode of dispersion in the latter medium will apply in almost
all points to the former: the principal difference consists in the
circumstance that in the horse-chestnut solution the dispersion
begins earlier in the spectrum than in the solution of quinine. In
a solution of sulphate of quinine of convenient strength, we have
seen that the dispersion came on at about G^H, the excessively
faint dispersion which was exhibited earlier being left out of con¬
sideration, whereas in a decoction of the bark of the horse-chestnut,
diluted so as to be of a convenient strength, it came on a little
before G. This explains the reason of an observation of Sir David
Brewster’s, who has remarked that “a beam of light that has
passed through the esculine solution disperses blue light, but not
copiously, when transmitted through the quinine solution ; but the
beam that has passed through quinine is copiously dispersed when
transmitted through esculine f.”

* [It is a glucoside.]

t Philosophical Magazine , Vol. xxxii. (June 1848), p. 406.

19—2

292

ON THE CHANGE OF REFRANGIBILITY OF LIGHT.

Green Fluor-Spar from Alston Moor .

32. It is well known that some specimens of fluor-spar exhibit
a sort of double colour. In particular, a variety found at Alston
Moor, which is green when seen by transmitted light, appears
when viewed in a certain manner of a beautiful deep blue. This
blue colour seems to have been considered by Sir John Herschel
as merely superficial. It has been shown however by Sir David
Brewster to arise from light dispersed in the interior of the crystal,
and to have no particular relation to the surface.

The crystal with which the following observations were made
was of a fine but not intense green when viewed by transmitted
light. On viewing a pure spectrum through it, there was found
to be a dark band of absorption in the red. This band was narrow,
and by no means intense. The crystal exhibited a copious deep
blue by dispersive reflexion.

33. On admitting into the crystal a cone of sunlight formed
by a lens of short focus, and then analysing the dispersed beam,
it was found to consist of a very little red followed by a dark
interval, then green, faintly fringed below with less refrangible
colours down perhaps to the orange, then blue, or bluish-green,
followed by a great deal of indigo or violet. Independently of the
gap in the red, the spectrum was not quite continuous, for a band
of bluish-green, not very broad, was separated by dusky bands
from the green below and the indigo above. The separate red
band and the two dusky bands were all so faint as to be difficult
to see.

The dispersed beam was readily proved to be truly dispersed,
for it was unpolarized, and a pale brown glass cut it off when
placed in the first position, although it transmitted it in a great
measure when placed in the second.

34. When the crystal was examined by the third method,
the general result closely resembled that produced by sulphate of
quinine. The dispersion commenced about half-way between G
and H , and continued from thence onwards far beyond H. It was
strongest about H. The fixed lines were seen with beautiful
distinctness as dark planes in the crystal. The groups H , Z, m
were quite evident, and n might be seen without difficulty. I have

ON THE CHANGE OF EE FRAN GIBILITY OF LIGHT.

293

even, seen some of the fixed lines of the groups The tint of the
dispersed light appeared as nearly as possible uniform throughout.
The distance to which this light could be traced from the surface
did not at all diminish so rapidly in this crystal, with an increase
in the refrangibility of the incident light, as it had done in the
case of a solution of sulphate of quinine. Indeed, it was difficult
to say how far the decrease in the depth to which the incident rays
could be traced, by means of the dispersed light which they pro¬
duced, was due merely to the increasing faintness of the light, and
how far it indicated a real increase in the absorbing energy of the
crystal; whereas in the case of sulphate of quinine the appearance
presented unequivocally indicated a very rapid increase of absorb¬
ing power.

35. On examining the crystal by the second method, the
general appearance was the same as in the case of sulphate of
quinine, but the beam of falsely dispersed light was absent. In
addition to the copious beam of deep blue light dispersed by the
most refrangible rays, there was however a faint beam of red or
reddish light dispersed by rays of low refrangibility. This beam
was too faint to be seen by the third method of examination. It
will be remembered that the prismatic analysis of the transmitted
light gave a band of absorption in the red. Another crystal of a
pale colour, which did not give a similar band of absorption in the
red, exhibited nothing but the blue beam of dispersed light when
examined by the second method.

3G. On examining the crystal by the fourth method, the
extreme red proved inactive. The activity commenced about the
most refrangible limit of the red transmitted by a deep blue glass,
when the dispersed light was red, but extremely faint. On moving
the lens onwards through the spectrum, the dispersed light rapidly
became brighter, and then died away. When at its brightest,
although even then it was almost too faint for prismatic examina¬
tion, it appeared to consist of not quite homogeneous light a little
lower in refrangibility than the active light. For a considerable
distance further on there was no sensible dispersion produced.
The dispersed light became again perceptible when the active
light belonged to the greenish yellow, or not till the blue, accord¬
ing to the intensity of the incident light. As the lens moved on

294 ON THE CHANGE OF REFRANGIBILITY OF LIGHT.

the dispersed light remained faint for a considerable time. It was
first reddish and then brownish, with a refrangibility answering to
its colour. When the active light was at G^H, or thereabouts,
the dispersed light rapidly grew much brighter, and became of a
fine blue. On analysis it was found to consist of rays the refran-
gibility of which ranged within wide limits. The red rays were,
however, almost wholly wanting, while the rays belonging to the
more refrangible part of the spectrum resulting from the analysis
of the dispersed beam were particularly copious. The most re¬
frangible limit of the dispersed light did not quite reach in
refrangibility the active light. The dispersed light was most
copious when the active light belonged to the neighbourhood of
H . As the lens moved on the dispersed light grew less bright,
and gradually died away.

Solution of Guaiacum in Alcohol.

37. This is one of the media mentioned by Sir David Brewster,
who remarks that it “ disperses, by the stratum chiefly near its
surface, a beautiful violet light/’

When this fluid is examined by the third or fourth method, it
is found to exhibit a copious internal dispersion, which begins to
be conspicuous much lower down in the spectrum than in the
cases already described. In observing by the third method, the
true dispersion appeared to commence about the end of the green,
the dispersed light being reddish-brown. By the fourth method
the dispersion could be traced as low down as DJ-&, the dispersed
light being reddish. As the lens moved onwards, in a direction
from the red to the violet, the more refrangible colours entered in
succession into the dispersed beam, and it became successively
brownish, yellowish, greenish, and bluish. In whatever part of
the spectrum the lens might be, it was found that the most
refrangible part of the dispersed beam was of lower refrangibility
than the active light. This could be easily determined by means
of the beam of falsely dispersed light, which was always visible
so long as the active light belonged to the visible part of the
spectrum.

38. With the third arrangement the fixed lines were seen as
before by means of the dispersed light, but in this fluid they

ON THE CHANGE OF REFRANGIBILITY OF LIGHT.

295

could be seen much lower down in the spectrum than in the
solution of sulphate of quinine. The group H was seen on a
greenish ground. About the group l the ground was still greenish,
but the dispersed light was not very copious. The beautiful violet
light mentioned by Sir David Brewster is produced only by rays
of extremely high refrangibility, and is found to extend from the
beginning of the group m to the end of the group n, and even
further. This part of the dispersion is best seen with a rather
dilute solution.

39. In a solution of guaiacum, just as in the solution of
sulphate of quinine, the absorbing power of the medium increases
very rapidly with the refrangibility of the light. This is shown
by the rapid decrease in the distance from the surface to which
the dispersed light can be traced. The reason why the violet
dispersed light is confined to a very thin stratum adjacent to the
surface by which the light enters, is simply that the medium is
so nearly opaque with regard to the invisible rays beyond the
extreme violet that all such rays are absorbed by the time the
light has passed through a very thin stratum of the fluid.

40. If the solution be strong the colour is of considerable
depth. In all such cases it is necessary to take the precaution,
mentioned by Sir David Brewster, of transmitting the incident
beam as near as possible to the upper surface, so as just to graze
it. The absorption of the medium would otherwise modify the
tint of the dispersed beam.

41. The solutions of quinine and guaiacum present a striking
contrast with respect to the change of tint of the dispersed beam.
In the former solution the change is but slight, if we except that
part of the dispersion which is very faint; whereas in the latter,
the prismatic colour which makes the nearest match to the com¬
posite tint of the dispersed beam runs through nearly the entire
spectrum, as the refrangibility of the active light changes from
that of the green rays to that of invisible rays situated far beyond
the extreme violet.

Tincture of Turmeric .

42. This fluid is very sensitive, and exhibits a pretty copious
dispersive reflexion of a greenish light. In its mode of internal

296

ON THE CHANGE OF REFRANGIBILITY OF LIGHT.

dispersion it strongly resembles a solution of guaiacum, but the
final tint of the dispersed light does not correspond to so high a
mean refrangibility. When the fluid was examined by the third
method, the true dispersion appeared to commence about b. The
absorbing power was so great for the rays of high refrangibility,
that from a little above F (in the case of tincture not diluted with
alcohol) to the end the dispersed light seemed to be confined to
the mere surface. By the fourth method the dispersion was as
usual traced a little lower down in the spectrum. When the
dispersed beam was first perceived it was nearly homogeneous,
and its refrangibility was only a very little less than that of the
active light. As the refrangibility of the active light increased,
new colours, in the order of their refrangibility, entered into the
dispersed beam, which became more and more composite, while at
the same time its upper limit became distinctly separated from
the beam of falsely dispersed light, which, when the whole dis¬
persed beam was analysed by a prism, was always found in
advance of the other. The tint of the dispersed beam passed
from orange through yellow to yellowish green, which was its
final tint. Tincture of turmeric is well adapted for exhibiting
the fixed lines in the invisible part of the spectrum, though per¬
haps not quite so well as a solution of sulphate of quinine.

Alcoholic Extract from the Seeds of the Datura Stramonium.

43. This fluid, which I was led to try in consequence of Sir
David Brewster’s paper, proved to be remarkably sensitive, and
exhibited a copious dispersive reflexion of a pale but lively green.
The general phenomena are so nearly the same as in a solution of
sulphate of quinine that there is no need of a separate descrip¬
tion. The principal difference consists in the tint, which is green
instead of blue. In the present case, however, the fluid, in addi¬
tion to its dispersion of green, dispersed a red beam under the
influence of certain red rays. As the lens employed in the fourth
method of examination was moved from the extreme red onwards,
the light was at first inactive, but when the lens reached a certain
point of the spectrum, a red beam of truly dispersed light sud¬
denly appeared, which disappeared with almost equal suddenness
as the lens moved on. In this mode of observation the refrangi¬
bility of the dispersed could hardly be distinguished from that of

ON THE CHANGE OF REFRANGIBILITY OF LIGHT. 297

the active light; but on combining the first and third methods,
by removing the lens, placing the vessel truly in focus, and hold¬
ing a blue glass alternately in front of the vessel and in front of
the eye, I satisfied myself that the truly dispersed beam, taken as
a whole, was of lower refrangibility than the light by which it was
produced. The utility of the blue glass depended upon the cir¬
cumstance that the upper extremity of the extreme red which it
transmitted nearly coincided with the point of the spectrum at
which the red beam occurred. This red beam was doubtless due
to the presence of a small quantity of chlorophyll, or one of its
modifications. The light transmitted by the fluid exhibited on
prismatic analysis the absorption band in the red which is so
characteristic of that substance.

The colour of the solution was a pale brownish yellow; it
would no doubt have been still paler, and perhaps nearly colour¬
less, had the sensitive principle to which the green dispersion was
due been present in equal quantity but in a state of purity. As
it was, the fluid was pale enough to exhibit well, when poured
into a test tube and held in front of a window, a narrow arc on
the side of the incident light, like sulphate of quinine, only in this
case the arc was green instead of blue.

Frequency of the occurrence of true internal dispersion having the
same general character as that which takes place in the cases
above described.

44. If we except the red dispersed beam produced by red
rays in the crystal of fluor-spar and in the stramonium extract, a
strong similarity may be observed in the mode of internal dis¬
persion which takes place in the cases hitherto described. As the
refrangibility of the incident light continually increases, the rays
are at first inactive. At a certain point of the spectrum, varying
according to circumstances, the true dispersion begins to be
sensible, but is faint at first. After remaining faint for some
distance it presently becomes more copious. It remains very
conspicuous through the whole of the violet and beyond, and then
gradually dies away. It consists at first of light of comparatively
low refrangibility, and then new colours in the order of their
refrangibility enter into it. Frequently the greater part of the

298

ON THE CHANGE OF RE FRA NGIBILIT Y OF LIGHT.

change of prismatic composition takes place while the dispersed
light is very faint, so that practically speaking we may almost
say that the tint is uniform. Sometimes, when the dispersion
just commences, the dispersed light is nearly homogeneous, and
has a refrangibility so nearly equal to that of the active light
that the beams due to true and false dispersion can hardly be
separated.

45. Now this, so far as I have observed, is much the com¬
monest kind of true internal dispersion, although sometimes the
phenomenon presents very striking singularities. In the paper
in which Sir David Brewster first announced the discovery of
internal dispersion, he remarks “ that it is a phenomenon which
occurs almost always in vegetable solutions, and almost never in
chemical ones or in coloured glasses*.” For my own part, I have
rarely met with a vegetable solution which did not exhibit more
or less the phenomenon of true internal dispersion. Its existence
may in general be easily detected in the following manner. The
sun’s light being reflected horizontally through a lens, a deep blue
glass is left in such a position as to intercept the light incident on
the vessel containing the fluid, which is placed at the focus of the
lens. A pale brown glass of the proper kind is then placed so as
to intercept, first the incident, and then the dispersed light. A
vessel with flat sides filled with a solution of sulphate of quinine
would be better, and then the placing of the medium in the
second position might be dispensed with, the medium being
sensibly transparent. Sometimes it is useful to have recourse to
analysis through a doubly refracting prism, or a rhomb of cal¬
careous spar. In this way true internal dispersion may often be
detected in a fluid which is actually muddy, in which case, were
the effect of the incident light observed as a whole, the true
would be masked by the enormous quantity of false dispersion
which such a medium would offer.

46. The fluids obtained by treating the leaves and other
parts of plants with alcohol or hot water are almost always sensi¬
tive, so far as I have observed. The solutions in water presently
ferment, and are frequently highly sensitive in the early stages of
fermentation; they are usually more or less sensitive in all stages.

Edinburgh Transactions , Yol. xn. p. 542.

ON THE CHANGE OF REFRANGIBILITY OF LIGHT.

299

Different kinds of fungus furnish very sensitive solutions. When
aqueous solutions become muddy by decomposition, other clear
and often highly sensitive liquids may be obtained from them by
various chemical processes. Port and sherry are decidedly sensi¬
tive. In such cases the fluid is a mixture of several substances,
of which some may be sensitive and others insensible. When
vegetable substances are isolated they are frequently insensible,
or else so very slightly sensitive when examined under great
concentration of the highly refrangible rays, that it is quite im¬
possible to say whether the sensibility thus exhibited may not be
due to some impurity: thus, several solutions containing sugar,
salicine, morphine, or strychnine were found to be insensible. A
solution of veratrine in alcohol proved to be sensitive in a pretty
high degree, dispersing internally a bluish light. Sir David
Brewster has remarked that a solution of sulphate of strychnine
in alcohol dispersed light after it had stood for some days. This
observation I have verified with reference to true dispersion,
which the solution exhibits, though not very copiously, after it
has been made some time. There can be little doubt that the
sensitive principle in this case is not strychnine, but some product
of its decomposition. I now come to some instances of internal
dispersion which are far more striking.

Solution of Leaf-Green in Alcohol.

47. It was in this very remarkable fluid that the phenomenon
of internal dispersion was first discovered by Sir David Brewster,
w r hile engaged in researches relating to absorption. The character
of the internal dispersion of a solution of leaf-green is no less
remarkable than the character of its absorption. On account of
the close connexion which seems to exist between the two phe¬
nomena, it will be requisite first to say a few words about the
latter.

When green leaves are treated with alcohol, a fluid is obtained
which is of a beautiful emerald-green in moderate thicknesses, but
red in great thicknesses, and which has a very remarkable effect
on the spectrum. A good number of the following observations
on the internal dispersion of leaf-green were made with a solution
obtained from the leaves of the common nettle, by first boiling
them in water and then treating them with cold alcohol, the

300

ON THE CHANGE OF REFRANGIBILITY OF LIGHT.

leaves having previously been partially dried by pressing them
between sheets of blotting paper. Nettle was chosen partly
because it stands boiling without losing its green colour, and
partly for other reasons. My object in boiling the leaves was to
obtain the green colouring matter more nearly in a state of isola¬
tion, but it seems to have the additional advantage of giving a
solution less liable to decomposition. Indeed, this fluid seemed
disposed to remain permanently unchanged when kept in the
dark; but a small portion of it which was exposed to strong light
had its colour rapidly discharged.

48. When fresh leaves are left in contact with alcohol in the
dark, or in only weak light, the colour of the fluid changes by
degrees, and it seems to approximate (making allowance for
impurities) to a type which is nearly represented by the fluid
obtained in this manner from laurel leaves, or that obtained by
treating with alcohol tea leaves from which a good deal of brown
colouring matter has first been extracted by water. This type
was rather ideal than actual, being derived from a comparison of
different cases, until it seemed to be realized in the case of a fluid
obtained by re-dissolving in alcohol a crust* which had formed
itself at the bottom of a test tube containing leaf-green. The
principle to which the peculiar absorption and internal dispersion
of such a fluid seems due may be called modified leaf-green. The
fluid itself is not green but olive-coloured, becoming red at great
thicknesses.

49. When solutions of leaf-green, and of its various modifica¬
tions, are examined in different thicknesses by the light of a
candle, there are five bands of absorption which may be observed

[* From the spectrum of its solution, this crust must have been the product of
decomposition by acids of the principal red-absorbing and red-fluorescing constituent
of the chlorophyll mixture. This mixture consists (in land plants) mainly of two
red-fluorescing and red-absorbing substances and a yellow non-fluorescent substance,
all showing characteristic bands of absorption. They are all three, especially the
first two, easily altered by acids. The “ modified leaf-green ” is the mixture with
the first two substances decomposed as if by the minute quantity of acid obtained
from the extract of the leaves. As the relative proportion of the three substances
in the extract is liable to vary with the circumstances of the manipulation, and as
the first two are very easily altered by the least trace of acid, there is some
uncertainty in endeavouring (as in § 49) to identify the bands of absorption
described by different observers who worked with the natural mixture.]

ON THE CHANGE OF REFRANGIBILITY OF LIGHT. 301

in the spectrum. These will be called, in the order of their
refrangibility, Nos. 1, 2, 3, 4 and 5, the bright bands below the
respective dark bands being also numbered in the same manner.
Of the dark bands, Nos. 1, 2, 3 and 5, are the first four in Sir
David Brewster s plate*. No. 4 is mentioned in the memoir, but
not represented in the plate, which corresponds to a thickness not
sufficient to bring out this band. The last band in the plate
could not be seen without strong light. The dark bands Nos. 1
and 2 are situated in the red, No. 3 about the yellow* or greenish
yellow, No. 4 in the green, and No. 5 early in the blue. Of these,
No. 1 is in small thicknesses by far the most intense, and it may
be readily seen even in a very dilute solution; it might apparently
be used as a chemical test of chlorophyll, or one of its modifications.
The test would be of very easy application, since it would be
sufficient to hold a test tube with the liquid at arm’s length before
a candle at a little distance, and view the linear image of the
flame through a prism applied to the eye.

50. Fresh and modified leaf-green differ much in the order in
which the bright bands are absorbed, and in the degree to which
the dark bands are developed before they cease to be visible by the
absorption of the part of the spectrum in which they are situated.
In the green fluid, the dark band No. 5 is not usually seen,
because the spectrum is there cut off, unless a very small thickness
be used. With a moderate thickness, Nos. 2 and 3, especially the
former, are well seen, and No. 1 is very intense. As the absorption
goes on, the bright bands Nos. 2 and 3 are absorbed, and there is
left the red band No. 1, and a double green band, consisting of the
bright bands Nos. 4 and 5, separated by the dark band No. 4,
which by this time has come out. In modified leaf-green, the dark
bands Nos. 4 and 5 are much more conspicuous than in the green
fluid, but No. 3 is wanting, or all but wanting. With a thickness
by which the absorption is well developed, the conspicuous bright
bands are in this case Nos. 1 and 3, and next to them No. 2,
whereas in the green fluid Nos. 2 and 3 were quickly absorbed, or
at least the whole of No. 2, and the greater part of No. 3.

51. It seems worthy of remark, that, especially in the case of
the green fluid, the absorbing power alters with the refrangibility

Edinburgh Transactions , Yol. xii.

302 ON THE CHANGE OF REFRANGIBILITY OF LIGHT.

of the light at a very different rate on the two sides of the intense
dark band No. 1. This might be inferred from the order in which
the bright bands disappear; but it was rendered visible to the eye
by the following easy experiment. A narrow test tube was partly
filled with a solution of leaf-green, and then a few drops of alcohol
were added, which remained at the top, and there diluted the
solution. The tube was then held before a candle, and the linear
image of the flame was viewed through a prism. In the under
part the dark band No. 1 was broad, the bright band No. 2 being
narrow, and almost obliterated, but in the upper part the dark
band No. 1 was very narrow. Now on tracing upwards the sides
of this dark band, it was found that the less refrangible side was
almost straight, and the diminution in the breadth of the band
was produced by the encroachment of the bright band No. 2.
Speaking approximately, we may say that in proceeding from the
extreme red onwards, at a certain point of the spectrum the fluid
passes abruptly from transparent to opaque, and then gradually
becomes almost transparent again.

52. It may here be remarked, that although the absorption
produced by leaf-green is best studied in a solution, its leading
characters may be observed very well by merely placing a green
leaf behind a slit, as near as possible to the flame of a candle, and
then viewing the slit through a prism.

53. After this digression relating to the absorption of leaf-
green, it is time to come to its internal dispersion. And first,
when a cone of white light coming from the sun is admitted
horizontally into the fluid, as close as possible to its upper surface,
and the beautiful red beam of dispersed light is analysed by a
prism, the spectrum is found to consist of a bright red band of a
certain breadth, followed by a dark interval, and then a much
broader green band not near so brilliant. There is usually but
little false dispersion, and what there is may be almost entirely
got rid of by analysing the beam by a Nicol’s prism, so as to view
it by light polarized in a plane perpendicular to the plane of
dispersion. Now on raising the vessel without removing the prism
from the eye, it was found that a dark band, which was in fact the
absorption band No. 1, appeared almost exactly in the middle of
the bright red band. On continuing to raise the vessel, so as to

ON THE CHANGE OF REFRANGIBILITY OF LIGHT. 303

make the dispersed rays pass through a still greater thickness of
the medium before reaching the eye, the dark band increased in
width, and when the red beam was almost absorbed, the part that
was left consisted of two cones of red, one at each side of the dark
band, which by this time had become broad. The whole appear¬
ance seemed to indicate that the bright red beam of dispersed
light had a very intimate connexion with the intense absorption
band No. 1.

54. Among coloured glasses, there is one combination which
produces a very striking effect. When a deep blue glass is placed
in the first position, the dispersed light, if the solution be at all
strong, is confined to a very thin stratum adjacent to the surface,
and is best seen by placing the vessel so that the surface of the
fluid at which the light enters is situated at a little distance on
either side of the focus of the lens, when there is seen a bright
circle of a most beautiful crimson colour. It might be supposed
that the red of which this circle mainly consists was nothing but
the extreme red transmitted by the blue glass. But it is readily
shown that such is not the case. For in the first place, the fluid
transmits pretty freely the red transmitted by the blue glass,
whereas the red light found in this circle is almost confined to the
surface of the fluid. Again, it was found that a pale brown glass,
which transmitted freely the extreme red, almost entirely cut off
the bright circle, when placed in the first position without removing
the blue glass, although it freely transmitted it when placed in
the second position. It appears, therefore, that the bright circle
is due, not to the red, but to the highly refrangible rays trans¬
mitted by the blue glass.

55. When a solution of leaf-green was examined by the third
method, the appearance as seen from the outside was very *
singular. The fixed lines in all the more refrangible part of the
spectrum were seen as interruptions in a bright red ground
verging to crimson. The beauty and purity of the tint, and the
strange contrast which it presented to the colours belonging to
that part of the spectrum, were very striking. About H the tint
began to verge towards brown, and the fixed lines beyond H were
seen on a brownish red ground. That the ground on which the
fixed lines of somewhat less refrangibility were seen was rather

304 ON THE CHANGE OF REFRANGIBILITY OF LIGHT.

crimson than red, arose, no doubt, from the mixture of a little
blue or violet light due to false dispersion, and to the scattering
which took place at the surface of the glass.

56. On looking down from above, the places of the more
conspicuous bands of absorption were indicated by dark teeth,
with their points turned towards the incident light, interrupting
the dispersed light. It is to be understood that the light was
transmitted as close as possible to the upper surface, so that the
absorption by which these teeth were formed took place before
dispersion. In this way the places of the absorption bands Nos.
1, 2 and 4, were perfectly evident. No. 3, it will be remembered,
was by no means conspicuous. When the solution is of convenient
strength, the absorption is so rapid beyond the bright band No. 5,
that the dispersion is confined to a thin stratum close to the
surface by which the light enters, and therefore no dark tooth
would be seen corresponding to the dark band No. 5.

57. On following the active light through the spectrum, in
the direction of increasing refrangibility, the dispersion was found
to commence with a bright but narrow tail of pure red light,
which shot right across the vessel. The light by which this tail
was produced belonged to the more refrangible part of the extreme
red band which is transmitted by a moderate thickness of the
fluid. The activity of the incident light commenced almost
abruptly: the same, it will be remembered, was the case with the
absorbing power of the medium. After the tail of red light came
the intense absorption band No. 1, where the dispersed light was
confined to the immediate neighbourhood of the surface by which
the active light entered. At this place a very bright band of
dispersed light was visible on looking at the vessel from the
outside. In this part of the spectrum the active and the dispersed
light were both red; but that dispersion was accompanied by a
change of refrangibility was shown by the effect of absorbing
media. Thus the long red tail and the bright band adjacent to
the surface were differently affected by a blue glass, according as
it was held in the first or the second position ; and the bright
band, though much enfeebled, was still plainly visible through a
considerable thickness of the fluid, although a stratum having a
thickness of only a very small fraction of an inch was sufficient to

ON THE CHANGE OF REFRANGIBILITY OF LIGHT.

305

absorb the rays by which the band was produced. Although the
dispersion continued throughout the whole of the visible spectrum
and beyond, it was comparatively feeble in the brightest part of
the spectrum. It became pretty copious again in the neighbour¬
hood of the dark band No. 4, and remained copious throughout
the blue and violet. In the green, the dispersed light was red,
slightly verging towards orange, and in the blue and violet it was
red verging a little towards brown.

58. It may seem superfluous, after what precedes, to bring
forward any further proof of the reality of a change of refrangi-
bility. Nevertheless the following experiment, which was in fact
performed at an early stage of these researches, may not be deemed
wholly unworthy of notice, as not involving the use either of
absorbing media or of false dispersion.

A small narrow triangle of white paper was stuck on to the
outside of the vessel containing the leaf-green, in such a manner
that its axis was vertical, and its vertex, which was uppermost,
was situated at the height of the middle of the spectrum. A
narrow vertical slit was then placed at the distance of the image
of the first slit, where the fixed lines were formed, and moved
sideways till the light immediately beside the fixed line 0 passed
through it. The vessel was then placed a few inches behind the
slit, and moved sideways till the riband-shaped beam of homo¬
geneous light, which passed through the second slit, was incident
on the vertex of the triangle. On looking at the vessel from the
front, as nearly as was convenient in the direction of the incident
light, there appeared a bright vertical bar corresponding to a
section of the incident beam. This bar was of two colours, namely,
red in the upper half, where the light fell on the fluid, and indigo
in the under half, where it fell on the paper. On refracting the
whole system sideways, through a prism of moderate angle
applied to the eye, the objects appeared in the following order as
regards refrangibility. First came the upper half of the bright
bar, which was only a very little widened by refraction, so that it
consisted of red light which was approximately homogeneous.
Next came the triangle, with its vertex a little rounded, and its
edges tinged with prismatic colours. The vertex, which had
formerly coincided with the bright bar, now lay a little to one

20

s. III.

306

ON THE CHANGE OF REFRANGIBILITY OF LIGHT.

side of its upper half. The triangle- was of course seen by means
of the diffused light of the room, which was not perfectly dark,
and therefore its refrangibility must have corresponded to the
brightest part of the spectrum, or nearly so. Lastly came the
under half of the bright bar, which was much more refracted than
the triangle, so as to be shifted almost completely off it. The
paper triangle was far too close to the first surface of the fluid to
allow of attributing the dislocation of the bright bar to any error
depending upon parallax; but to prevent all possible doubts on
this score, I took care to refract the system both right and left,
and the result was the same in the two cases. The conclusion is
therefore inevitable, that the indigo light which had changed its
colour by dispersion from leaf-green had changed its refrangibility
at the same time.

59. In viewing a solution of leaf-green in a pure spectrum, I
•noticed a phenomenon which further indicates the close connexion
which seems to exist between the absorption and internal disper¬
sion of this fluid. On holding the eye vertically over the fluid,
and looking down at the dispersed light through a red glass, I
observed five minima of illumination, having for the most part the
shape of teeth with their bases situated at the surface by which
the light entered, and their points turned inwards. These minima
occupied positions intermediate between the bands of absorption,
so far at least as the positions of the latter were indicated by dark
teeth pointing in the contrary direction. The first minimum was
situated a little beyond the intense absorption band No. 1, and
corresponded in position to the bright band No. 2. The second
was situated a little further on. The maximum intervening
between this and the third was but slight, so that the second and
third together formed pretty nearly one broad minimum. The
third and fourth were situated one at each side of the dark band
No. 4, so as to correspond in position to the bright bands Nos. 4
and 5. The fifth was situated a little way beyond the bright
band No. 5. This last minimum was not tooth-shaped, inasmuch
as it occurred at a part of the spectrum where the dispersed light
was almost confined to the surface of the fluid. These minima
are best seen when the solution is rather weak. They may be
perceived without using a red glass, though not so easily as with
its assistance. With a stronger solution it was observed that the

ON THE CHANGE OF REFRANGIBILITY OF LIGHT. 307

first minimum ran obliquely into the dark tooth corresponding to
the absorption band No. 1.

60. The reason of the occurrence of these minima appears
to be simply this, that the more copiously dispersed light is
produced, the more rapidly the incident light is used up in pro¬
ducing it, so that minima of activity correspond to points of the
spectrum at which the incident light penetrates to comparatively
great distances into the fluid before it is absorbed. The oblique
position observed in the first minimum is readily explained by
considering that the illumination at any point of the field of
view depends conjointly upon the activity of the incident light,
which is a function of its refrangibility, and upon the fraction of
the incident light left unabsorbed, which last is a function both of
the refrangibility and of the distance from the first surface.

61. It seems worthy of remark, that while the quantity of
dispersed light is liable to fluctuations having an evident relation
to the bands of absorption which occur throughout the spectrum,
the quality of the light dispersed, as regards its refrangibility,
appears rather to have reference to the intense absorption band
No. 1.

Extract from blue leaves of the Mercurialis perennis.

62. The juice of this plant has the property of turning blue
by exposure to the air. Some leaves and stalks which had
turned blue were treated with alcohol, and a green fluid was thus
obtained much resembling in colour the ordinary solutions of leaf-
green, but I think of a rather bluer green than usual. In its
mode of absorption, too, it much resembled ordinary solutions of
leaf-green, to which substance no doubt the greater part of its
colour was due. Its internal dispersion however was very
peculiar, for it dispersed a copious orange in place of a blood red
like the extracts from fresh green leaves in general, those of the
Mercurialis perennis included. On analysis the dispersed beam
was found to consist chiefly of a red band, similar to that which
occurs in solutions of leaf-green, and of a yellow or orange and
yellow band, a good deal brighter than the former, from which it
was separated by an intervening dark band. When the fluid was

20—2

308

ON THE CHANGE OF REFR ANGIBILITY OF LIGHT.

examined by the second method, it was found that the yellow
dispersion was produced principally by the brightest part of the
spectrum. After a considerable time the fluid lost its fine green
colour, as is very often the case with solutions of leaf-green, and
became yellowish brown, but the red and yellow dispersions still
continued.

When the fluid was examined by the fourth method, it was
found that the red rays dispersed a red, just as in a solution
of leaf-green. The additional dispersion which was so con¬
spicuous in this fluid began almost abruptly about the fixed line
JD. When it was first observed, the refrangibility of the orange
dispersed light could hardly, if at all, be separated from that of
the active light. As the lens moved on, the orange beam rapidly
grew brighter, and yellow entered into it; and now it was easy to
see that the beam of falsely dispersed light lay at its more re¬
frangible limit. The orange and yellow dispersed beam was
brightest at about D§ E ; but though it decreased in intensity
it could be traced far beyond that point, in fact, throughout the
spectrum.

63. I have generally found that when a copious dispersion
commences almost abruptly at a certain point of the spectrum, it
is followed by a band of absorption in the transmitted light.
This law did not seem applicable to the orange dispersion ex¬
hibited by the solution just mentioned; but then it is to be
remembered that the solution contained a quantity of chlorophyll,
which produces absorption bands with such energy that it would
naturally mask the bands which might be due to another
colouring principle with which it was mixed. To try whether
the law would be obeyed if the chlorophyll were got rid of, I
boiled in water some portions of the root and young shoots
which had turned blue, chlorophyll being insoluble in water.
The solution thus obtained was red, in small thicknesses pink,
and dispersed copiously a yellow or rather orange light. On
subjecting the fluid to prismatic analysis, a band of absorption
was seen at the place expected. Since aqueous solutions of this
nature are liable to decomposition, frequently decomposing before
sunlight can be obtained by which to examine them, the red
solution was concentrated by evaporation and purified by alcohol,
in which the orange-dispersing principle is soluble, as had already

ON THE CHANGE OF REFRANGIBILITY OF LIGHT.

309

appeared from the properties of the alcoholic solution. The
alcoholic solution thus obtained remained unchanged, at least for
a long time, and had the further advantage over the aqueous
solution of presenting the sensitive principle more nearly in a
state of isolation, though it was still contaminated by some
principle which dispersed a whitish light under the influence of
rays of high refrangibility.

64. The blue colouring matter may be readily extracted by
cold water, but is decomposed by boiling. The blue solution dis¬
persed an orange light like the other, but the dispersed light
could not be nearly so well seen, just as would be the case were
the red orange-dispersing fluid mixed with an insensible blue
fluid of a much deeper colour, so that the mixture of the two
would be blue. And in fact when the blue fluid was changed to
red by boiling the colour became far less intense.

Archil and Litmus .

65. It is stated by Sir David Brewster that a very remark¬
able example of internal dispersion, which had been pointed out
to him by Mr Schunck, is exhibited in an alkaline or in an alco¬
holic solution of a resinous powder produced from orcine by con¬
tact with the oxygen of the air. Not being able readily to
procure a specimen of orcine, I tried archil, and obtained from
it and litmus some very remarkable solutions.

In the fluid state in which archil is sold, the colour is much
too deep for convenient optical examination. When a small
quantity of archil is diluted with a great deal of water, the
diluted fluid is very sensitive. It is red by transmission, or in
small thicknesses purple, but exhibits by dispersive reflexion a
pretty copious but rather sombre green.

66. When the fluid was examined by different methods, it
was found to disperse a little red, some orange, and a great deal
of green. The red dispersion was so slight, that in observing by
the third method it appeared doubtful whether there was any
except false dispersion. It commenced in the red, when the
active and dispersed lights had the same refrangibility, or nearly
so. The orange dispersion commenced about the fixed line D,

310 ON THE CHANGE OF REFRANGIBILITY OF LIGHT.

the dispersed light being at first nearly homogeneous, and of the
same refrangibility as the active light. On proceeding onwards
in the spectrum, in observing by the fourth method, the orange
beam became brighter, and yellow entered into it, but no colour
beyond that, so that the orange and yellow beam was left behind
by the beam of falsely dispersed light, from which it was separated
by a perfectly dark interval. The green dispersion began about b,
or a little beyond, coming on almost abruptly. The manner of its
commencement was best observed by the fourth method, by
holding a prism to the eye while the lens was moved through the
spectrum. In this way it was found that on arriving at the point
of the spectrum above mentioned, a gleam of green light shot
across the dark space which before separated the beam of falsely
dispersed light from the orange beam of truly dispersed light.
As the lens moved on, the green dispersed light grew brighter,
but its more refrangible limit did not seem to pass, or at least
much to pass, the refrangibility it had at first; so that the green
beam of truly dispersed light was almost immediately left behind
by the beam of falsely dispersed light. The former, on being left
behind, soon died away.

67. We might suppose either that the red, orange and
green dispersions are due to the same sensitive principle, or that
they are produced by three distinct sensitive principles mixed
together in the solution. The latter would appear the more
probable supposition, to judge by the apparent want of connexion
between the three dispersions. This view is strongly confirmed
by the following results. Some ether was poured on archil in the
fluid state, and after being gently moved about and allowed to
stand, a little was withdrawn without agitation. A purplish rose-
coloured fluid was thus obtained, which was highly sensitive,
exhibiting the orange and green dispersions but not the red.
The orange dispersion was far more copious, in proportion to the
whole quantity of dispersed light, than had been the case with
archil diluted with water.

Some archil was violently agitated with ether, and after
subsidence the ether was withdrawn. This ethereal solution
was much deeper in colour than the former, and exhibited the
red dispersion in addition to the orange and green. On adding a
small quantity of water, and agitating, a separation, or at least

ON THE CHANGE OF KEFRANGIBILITY OF LIGHT.

311

partial separation, of the sensitive principles took place; for the
upper fluid exhibited the orange dispersion abundantly, but none
of the red, and little or none of the green, while the under fluid
exhibited the green and red dispersions with little, if any, of the
orange. The upper fluid exhibited a pretty copious dispersive
reflexion of reddish orange, and the under fluid a remarkably
copious reflexion of a fine green. A similar separation, more or
less perfect, took place in other cases, the dispersion of orange
bearing to that of green a greater ratio in the ether than in the
water. Some of the green-dispersing fluids thus obtained were
most remarkable on account of the extraordinary copiousness of
the reflected green, and the strange contrast which it presented
to the transmitted tint, which was a purplish red.

The red dispersion in the second ethereal solution, though
decided, was by no means copious. In the case of archil merely
diluted with water, it had been so slight that its existence might
have been considered doubtful. It might be supposed that the
first solution was not sufficiently concentrated to exhibit the red
dispersion, in which case the red and green dispersions might have
been due to the same sensitive principle. But an ethereal extract
from dried archil, which was plainly concentrated enough, did not
exhibit the red dispersion, although it did exhibit the orange and
green dispersions. None of the sensitive principles appear to
constitute the chief part of the colouring matter of this dye-stuff.

68. When some of these ethereal solutions were examined by
the third method, with a lens of shorter focus than usual, the
appearance was very singular. At the less refrangible end of the
spectrum the incident light was quite inactive; and then, on
reaching a certain point, a copious dispersion of orange commenced
abruptly. This continued with no particular change for some
distance further on, when it passed abruptly into green. The
fourth method showed however that the former dispersion con¬
tinued, and was only masked, in the third method of observation,
by a new and more powerful dispersion of green which then
commenced. And in fact when the green-dispersing principle
was separated, or partially separated, by water, the orange dis¬
persion was seen to continue where before it appeared to have
been exchanged for green.

312 ON THE CHANGE OF KEFBANGIBILITY OF LIGHT.

69. I ought here to mention that a similar separation did not
take place on the addition of water only to an ethereal extract
from archil previously dried. The condition which determined
the separation in the first case appeared to be the presence of a
small quantity of ammonia, which would evaporate on drying the
archil. And in fact when a small quantity of ammonia was added
to the extract from dried archil, a partial separation was effected.

I do not here enter into the question whether one of the sensitive
principles may be obtained from the other, whether, for example,
a chemical combination of the orange-dispersing principle with
ammonia might disperse a green, or a green with a little orange.
A solution containing a mixture of the same substance in two
different states of chemical combination, both compounds being
sensitive, is not the less justly regarded as containing two distinct
sensitive principles.

70. The preceding results are mentioned, not for their own
sake, but merely for the sake of the method of examination em¬
ployed. The results indeed are so imperfect as to be worthless on
their own account. A complete optico-chemical examination of
archil and litmus would itself alone furnish a subject for research
of no small extent; but it belongs rather to chemistry than to
general physics. It is quite possible that internal dispersion may
turn out of importance as a chemical test. The dispersing such a
tint, and the having the dispersed light produced by light of such
a refrangibility, form together a double character of so peculiar a
nature that it enables us, so to speak, to see a sensitive principle
in a solution containing many substances, some of them, perhaps,
coloured, so that the colour of the solution may be very different
from what it would be if the sensitive principle were present alone.

71. The law mentioned at the beginning of Art. 63 did not
seem very applicable to archil when the fluid was merely diluted
with water. But when the orange-dispersing and green-dispersing
principles were obtained, as it would appear, more nearly in a
state of isolation, by means of ether and water, the law was found
to be obeyed. Thus, when the ethereal solution which exhibited
the orange dispersion and little else was examined by the third
method, the dispersion was found to commence with a tail of light
followed by a dark tooth, indicating the position of a band of

ON' THE CHANGE OF REFRANGIBILITY OF LIGHT.

313

absorption. When the light transmitted by a certain thickness of
this fluid was subjected to prismatic examination, it was found to
consist of red followed by some orange, when the spectrum was
cut off with unusual abruptness. After a broad dark interval
came the most refrangible colours faintly appearing. Those
solutions which exhibited a copious dispersion of green gave, in
addition to a band obliterating the yellow, a very distinct hand
separating the green from the blue. A similar band, but by no
means distinct, might be seen in archil merely diluted; and it is
particularly to be observed that this band, which occurred a little
above the point of the spectrum where the green dispersion com¬
menced, became more conspicuous when the green-dispersing
principle was present more nearly in a state of isolation.

72. Two portions of litmus were treated, one with ether and
the other with alcohol, which were allowed to remain in contact
with the solid. Both extracts, but especially the latter, were
highly sensitive, exhibiting dispersions of orange and green similar
to archil, and due apparently to the same sensitive principles.

The ethereal extract dispersed chiefly orange, while the alcoholic
extract dispersed orange and green in nearly equal quantities.

The latter extract exhibited a remarkably copious dispersive
reflection of a colour nearly that of mud, and was altogether one
of the strangest looking fluids that I have met with. On viewing
it in such a manner that no transmitted light entered the eye,
one might almost have supposed that it was muddy water taken
from a pool on a road. But when the bottle containing it was
held between the eye and a window the fluid was found to be
perfectly clear, and of a beautiful purple colour.

Canary Glass.

73. Among media which possess the property of internal
dispersion in a high degree, Sir David Brewster mentions a yellow
Bohemian glass, which dispersed a brilliant green light. This led
me to seek for such a glass, and it proved to be pretty common in
ornamental bottles and other articles. The colour of the glass by
transmitted light is a pale yellow. Its ornamental character
depends in a great measure upon the internal dispersion, which
occasions a beautiful and unusual appearance in the articles made

CARNEGIE INSTITUTE

314 ON THE CHANGE OF REFRANGIBILITY OF LIGHT.

of it. The commercial name of the glass is canary glass. The
following observations were made with a small bottle of English
manufacture.

74. When the sun’s light was admitted without decomposition
the dispersed beam was yellowish green. The 1 dispersion was so
copious that when a large lens was used the dispersed beam
approached to dazzling. The prismatic composition of this beam
was extremely remarkable. The beam was found on analysis to
consist of five bright bands, which were equal in breadth and
equidistant, or at least very nearly so, and were separated by
narrow dark bands. The first bright band was red, the second
reddish orange, the third yellowish green, the fourth and fifth
green. I have very frequently observed dark bands, or at least
minima, in the spectrum resulting from the prismatic analysis of
dispersed beams, but I have not met with any example so re¬
markable as this, except in a class of compounds which the
properties of canary glass led me to examine.

75. On analysing a beam of sunlight transmitted through a
certain thickness of the glass, there was found to be a dusky
absorption band a little below F, another less distinct at F\G,
and the spectrum was cut off a little below G .

76. When the glass was examined by the third method, the
dispersion was found to commence abruptly about the fixed line b.
It remained remarkably copious throughout the whole of the
visible spectrum and far beyond, with the exception of a band
beginning a little above F, and having its centre at about F^G,
where there was a remarkable minimum of activity. This band,
it will be observed, was situated between the bands of absorption
already mentioned. The tint of the dispersed light appeared to
be uniform throughout, except perhaps where the dispersion was
just commencing. This was the best medium I have met with for
showing the fixed lines of extreme refrangibility, though some
others were nearly as good.

77. On examining the glass by the fourth method, it was
found that the dispersion commenced nearly where the dispersed
light ended, that is, the lowest refrangibility of the rays capable of
being dispersed was nearly the same as the highest refrangibility

ON THE CHANGE OF KEFKANGIBILITY OF LIGHT.

315

of the rays constituting the dispersed beam exhibited by white
light as a whole. The dispersion appeared indeed to commence a
little earlier, at about the refrangibility of the fourth dark band
in the spectrum of the entire dispersed beam. When the small
prism was held to the eye with one hand, while the small lens in
the board was gradually moved with the other, in a direction from
the red to the violet, through the part of the spectrum where the
dispersion commenced, it was found that the region of the first
four bands was lighted up almost simultaneously, the whole field
of view having been previously dark. When the lens was moved
a very little further on the dispersed beam with its five bands
was formed complete. Indeed the whole five appeared almost
simultaneously. Speaking approximately, and in fact with almost
perfect accuracy, we may say that if white light be conceived to
be decomposed into two portions, the first containing rays of all
refrangibilities up to that of the fixed line b , or thereabouts, and
the second containing rays of all greater refrangibilities, the
dispersed light produced by white light as a whole belongs ex¬
clusively to the first portion; and yet, were the bottle illuminated
by the first portion alone, no dispersion whatsoever would be
produced, whereas were it illuminated by the second portion alone,
which contains not a ray having the same refrangibility as any
one of the dispersed rays, the dispersion would be exhibited in full
perfection.

Common Colourless Glasses.

78. Sir David Brewster states that he has met with many
specimens, both of colourless plate and colourless flint glasses, which
disperse a beautiful green light. All the colourless glasses which
I have examined dispersed light internally to a greater or less
extent, with the exception of some few specimens belonging to
Dr Faraday’s experiments. A beautiful green seems to be the
commonest tint of the dispersed beam, and this I have found in
wine glasses, decanters, apothecaries’ bottles, pieces of unannealed
glass, &c.; also in many specimens of plate and crown glass. The
green was generally of a finer tint than that dispersed by the
canary glass, but was not near so copious. On analysis it w T as
found to consist usually of red and green separated by a dark
band, or rather a minimum of brightness. Those specimens

316 ON THE CHANGE OF REFRANGIBILITY OF LIGHT.

which were examined by the third and fourth methods were
found to exhibit a little false dispersion, produced chiefly in the
brightest part of the spectrum, but the greater part was true
dispersion. This dispersion was produced chiefly by a rather
narrow band, comprising the fixed line 0, where there appeared
to be a remarkable maximum of sensibility. The line G lay a
little above the lower limit of the band. Below the band dis¬
persion also took place, though not near so copiously, and there
appeared to be another maximum of sensibility some way further
down in the spectrum; but above the band dispersion almost
entirely ceased of a sudden; a very unusual circumstance when
the active and the dispersed light are well separated in re-
frangibility. The position of the band in the spectrum, and the
distribution of the illumination in it, which are very peculiar,
were the same in all the specimens which were sufficiently
sensitive to admit of being examined by the third method, but the
tint of the dispersed light was not quite the same.

79. Orange-coloured glasses are frequently met with which
reflect from one side, or rather scatter in all directions, a copious
light of a bluish-green colour, quite different from the transmitted
tint. In such cases the body of the glass is colourless, and the
colouring matter is contained in a very thin layer on one face of
the plate. The bluish-green tint is seen when the colourless face
is next the eye. As this phenomenon was supposed by Sir John
Herschel to offer some analogy with the reflected tints of fluor¬
spar and a solution of sulphate of quinine, I was the more desirous
of determining the nature of the dispersion. It proved on exam¬
ination to be nothing but false dispersion, so that the appearance
might be conceived to be produced by an excessively fine bluish-
green powder contained in a clear orange stratum, or in the
colourless part of the glass immediately contiguous to the coloured
stratum. The phenomenon has therefore no relation to the tints
of fluor-spar or sulphate of quinine. It is true that the very
same glass which displayed a superficial reflexion of bluish green,
when examined by condensed sunlight exhibited also, in its
colourless part, a little true dispersion, just as another colourless
glass would do. But this has plainly nothing to do with the
peculiar reflexion which attracts notice in such a glass.

ON THE CHANGE OF REFRANGIBILITY OF LIGHT.

317

Observations on the preceding results .

80. There is one law relating to internal dispersion which
appears to be universal, namely, that when the refrangibilitv of
light is changed by dispersion it is always lowered. I have
examined a great many media besides those which have been
mentioned, and I have not met with a single exception to this
rule. Once or twice, in observing by the fourth method, there
appeared at first sight to be some dispersed light produced when
the small lens was placed beyond the extreme red. But on
further examination I satisfied myself that this was due merely to
the light scattered at the surfaces of the large prisms and lens,
which thus acted the part of a self-luminous body, emitting a
light of sufficient intensity to affect a very sensitive medium.

81. Consider light of given refrangibility incident on a given
medium. Let some numerical quantity be taken for a measure of
the refrangibility, suppose the refractive index in some standard
substance. Let the refrangibilities of the incident and dispersed
light be laid down along a straight line AX (fig. 2) taken for the
axis of abscissae; let AM represent the refrangibility of the inci¬
dent light, and draw a curve of which the ordinates shall represent
the intensities of the component parts of the truly dispersed beam.
According to the law above stated, no part of the curve is ever
found to the right of the point M\ but in other respects its form
admits of great latitude. Sometimes the curve progresses with
tolerable uniformity, sometimes it presents several maxima and
minima, or even appears to consist of distinct portions. Some¬
times it is well separated from M, as in fig. 2; sometimes it
approaches so near to M that the most refrangible portion of the
truly dispersed beam is confounded with the beam due to false
dispersion.

82. Let f(x) be the ordinate of the curve corresponding to
the abscissa x, a the abscissa of the point M. Since f(x) is equal
to zero when x exceeds a, the curve must reach the axis at the
point M at latest, unless we suppose the function capable of alter¬
ing abruptly, as is represented in fig. 3. I do not think that such
an abrupt alteration, properly understood, is necessarily in contra-

318

ON THE CHANGE OF EEFRANGIBILITY OF LIGHT.

diction with the law of continuity. For the sake of illustration,
let us consider the phenomenon of total internal reflexion. Let P
be a point in air situated at the distance z from an infinite plane
separating air from glass. Conceive light having an intensity
equal to unity, and coming from an infinitely distant point, to be
incident internally on this plane at an angle 7 + 0 , where 7 is the
angle of total internal reflexion. The intensity at P is commonly,
and for most purposes correctly, considered as altering abruptly
with 6 , having, so long as <9 is negative, a finite value which does
not vanish with 8 , but being equal to zero when 8 is positive.
The mode in which the law of continuity is in this case obeyed is
worthy of notice. In the analytical expression for the vibration,
when 8 passes from negative to positive, the coordinate z passes
from under a circular function into an exponential with a negative
index, containing in its denominator A, the length of a wave of
light. As 6 increases through zero, the expression for the vibra¬
tion alters continuously; but if 0 be large compared with X it
decreases with extreme rapidity when 8 becomes positive. On
account of the excessive smallness of X, it is sufficient for most
purposes to consider the intensity as a function of 8 which vanishes
abruptly; and indeed it would be hardly correct to consider it
otherwise. For the use of the term intensity implies that we are
considering light as usual, whereas those phenomena which require
us to take into account the disturbance in the second medium
which exists when the angle of incidence exceeds that of total
internal reflexion, lead us to consider the nature as well as the
magnitude of that disturbance, which no longer consists of a series
of plane waves constituting light as usual. It is in some similar
sense that I mean to say that we may suppose the function f {x),
which expresses the intensity of the truly dispersed light, to alter
abruptly, without thereby implying any violation in the law of
continuity. In observing by the fourth method, the portion of the
spectrum operated on, though it may be small, is necessarily finite,
and in some cases no separation could be made out between the
beams of truly and falsely dispersed light. Hence I cannot under¬
take to say from observation, whether the variation of f (x) be
always continuous, though sometimes very rapid, or be in some
cases actually abrupt. I think, however, that observation rather
favours the former supposition, a supposition which, independently
of observation, seems by far the more likely.

ON THE CHANGE OF REFRANGIBILITY OF LIGHT.

319

83. Although the law mentioned in Art. 80 is the only one
which I have been able to discover, relating to the connexion
between the intensity and the refrangibility of the component
parts of the dispersed beam, which appears to be always obeyed,
and which admits of mathematical expression, there are some
other circumstances usually attending the phenomenon which
deserve notice.

When dispersion commences almost abruptly on arriving at
a certain point of the spectrum, the dispersed beam is very
frequently almost homogeneous at first, and of the same refrangi¬
bility as the active light. If the dispersed beam, when first
perceived, be decidedly heterogeneous, its refrangibility extends
almost, if not quite, to that of the active light, so that it is diffi¬
cult, if not impossible, to separate the beams of truly and falsely
dispersed light. On the other hand, when dispersion comes on
gradually, it is generally found that the refrangibility of even the
most refrangible part of the dispersed beam does not come up to
that of the active light.

Thus in the cases of the red dispersion exhibited by a solution
of leaf-green, and of the orange dispersions exhibited by solutions
obtained from archil and from the Mercurialis perennis, the dis¬
persed light was at first nearly homogeneous, and of the same
refrangibility as the active light. In the case of the green disper¬
sions shown by a solution obtained from archil, and by canary
glass, the dispersed light was heterogeneous from the first; but
still, when it first commenced, a portion of it had nearly the same
refrangibility as the active light. In a solution of sulphate of
quinine the dispersion came on gradually, being perceptible when
the active light belonged to the middle of the spectrum; and in
this case the dispersed light consisted of colours of low refrangi¬
bility. The bright part of the dispersion however came on pretty
rapidly, when the active light approached the extreme limit of the
visible spectrum, and accordingly the dispersed beam consisted in
that case chiefly of light of high refrangibility.

84. The mode of absorption of any medium may very con¬
veniently be represented by a curve, as has been done by Sir John
Herschel. To represent geometrically in a similar manner the
mode of internal dispersion, would require a curved surface. Let

320

ON THE CHANGE OF REFRANGIBILITY OF LIGHT.

the refrangibility of light be measured as before, and suppose for
simplicity’s sake the intensity of the incident light to be indepen¬
dent of the refrangibility, so that dy may be taken to represent the
quantity of incident light of which the refrangibility lies between
y and y + dy. Considering the effect of this portion of the incident
light by itself, let x be the refrangibility of any portion of the dis¬
persed light, and zdxdy the quantity of dispersed light of which
the refrangibility lies between x and x + dx. Then the curved
surface, of which the coordinates are x, y , a, will represent the
nature of the internal dispersion of the medium. We must sup¬
pose the intensity of the incident light referred to some standard
independent of the eye, since the illuminating power of the rays
beyond the violet, and even of the extreme violet, is utterly
disproportionate to the effect which in these phenomena they
produce.

From the nature of the case, the ordinate z of the surface can
never be negative. The law mentioned in Art. 80 may be ex¬
pressed by saying, that if we draw through the axis of z a plane
bisecting the angle between the axes of x and ?/, at all points
on the side of this plane towards x positive, the curved surface
confounds itself with the plane of xy.

85. Let us consider the form of this surface in two or three
instances of internal dispersion. For facility of explanation, sup¬
pose the plane of xy horizontal, let ./• be measured to the right, y
forwards, and 5 upwards. Let a line drawn in the plane of xy
through the origin, and bisecting the angle between the axes of x
and ?/, be called for shortness the line L, hi all cases the surface
rises above the plane of xy only to the left of the line L.

In the case of a solution of leaf-green, the surface consists as it
were of two mountain ranges running in a direction parallel to the
axis of y, or nearly so. The first range, if prolonged, would meet
the axis of x at a point corresponding to the place of the dark
hand No. 1 in the red, or nearly so. The second would meet it
somewhere in the place corresponding to the green. The green
range is much broader than tin*, red, but very much lower, and is
comparatively insignificant. The ridge of the rod range is by no
moans uniform, but presents a succession of maxima and minima.
The range commences at the end nearest to the axis of x with a
very high peak, by far the highest in the whole surface. In

ON THE CHANGE OF REFRANGIBILITY OF LIGHT. 321

following the ridge forwards, five minima or passes may be observed,
with hills intervening. The ordinates y of the first four of these
minima correspond to the refrangibilities of the bright bands Nos.
2 , 3, 4 and 5. The last minimum lies a little further on. Whether
similar minima exist in the green range is not decided by observa¬
tion, on account of the faintness of the green dispersed light.

In the case of canary glass, the surface consists of five portions
like mountain ranges running parallel to the axis of y , and having
abscissae belonging to the red, reddish orange, yellowish green,
green, and more refrangible green, respectively. These ranges do
not all start from the immediate neighbourhood of the line L , but
on the side towards the axis of x end almost in cliffs, at points at
which the ordinate y is nearly equal to the abscissa of the fifth
range, perhaps a little less. Thus the first three ranges are well
separated from the line L. The ranges are intersected by a sort of
valley running parallel to the axis of x } and having for its ordinate
y the refrangibility of F^G. With the exception of the minima
which occur where the ranges are intersected by this valley, the
ridges run on very uniformly, and it is only very gradually that
the ranges die away.

The form of the surface which expresses the internal dispersion
of a solution of sulphate of quinine, may be gathered from the
description of that medium. In this case the surface resembles a
rising country, not intersected by any remarkable mountain ranges
or valleys.

Fig. 4 is a rude representation of the internal dispersion in a
solution of leaf-green. The curves represented in the figure must
be supposed to be turned through 90° about the lines on which
they stand, and will then represent sections of the surface already
described, made by vertical planes parallel to the axis of x. OL is
the straight line bisecting the angle xOy. The figure is merely
intended to assist the reader in forming a clear conception of the
general nature of the phenomena, and must not be trusted for
details. No attempt is made to represent the several maxima and
minima in the intensity of the red beam of dispersed light. In
any such figure, if we suppose homogeneous light to be incident
on the medium, and wish to lay down the place of the falsely dis¬
persed beam, we have only to draw a straight line parallel to the
axis of x y through the point in the axis of y which corresponds to

21

s. III.

322 ON THE CHANGE OF REFRANGIBILITY OF LIGHT.

the refrangibility of the incident light, and find where this line
cuts the straight line OL which bisects the angle xOy.

On the cause of the clearness of fluids , notwithstanding a copious
internal dispersion which they may exhibit.

86 . It has been already remarked, that though water holding
a water colour in suspension makes an admirable imitation of a
highly sensitive fluid, when the latter is viewed by dispersive
reflexion alone, the two fluids have a totally different appearance
when viewed by transmitted light. The cause of this difference
appears to be plain enough. The light due to internal dispersion
emanates from each portion of the fluid which is under the influ¬
ence of the active light, and emanates apparently in all directions
alike. I have not attempted to determine experimentally whether
the intensity is strictly the same in all directions. The experiment
would be very difficult, especially for directions nearly coinciding
with that of the active light, because in that case the light which
was really due to internal dispersion would be mixed up with the
glare which is always found in the neighbourhood of light of
dazzling brightness. However, I have seen nothing which led me
to suppose that the intensity was different in different directions.
We may express the results of observation (extremely well, by
saying that the fluid or solid medium is self-luminous so long as
it is under the influence of the active light.

Accordingly, when a bright object, such as the sky, or the
flame of a candle, is viewed through a highly sensitive fluid, the
regularly transmitted light is accompanied by some side light
due to internal dispersion. The latter, however, emanating in
all directions alike from the influenced particles, is too faint,
when contrasted with the regularly transmitted light, to make
any sensible impression on the eye. But when a fluid, itself
insensible, holds in suspension a great number of solid particles
of finite size, the light reflected from such particies is rein forced,
in directions nearly coinciding with that of the incident light, by
a great quantity of diffracted light, so that a bright object viewed
through such a fluid is surrounded by a sort of nebulous haze,
giving the fluid a milky appearance.

ON THE CHANGE OF REFRANGIBILITY OF LIGHT.

323

Washed Papers.

87. In a paper “On the Action of the Eays of the Solar
Spectrum on Vegetable Colours,” Sir John Herschel mentions
a peculiarity which he had observed in paper washed with tinc¬
ture of turmeric, which consists in its being illuminated, when
a pure spectrum is thrown on it, to a much greater distance at
the violet end than is the case with mere white paper* This
phenomenon was attributed by Sir John to a peculiarity in its
reflecting power, and was considered as a proof of the visibility
of the ultra-violet rays. The colour of the prolongation of the
spectrum was yellowish green. Sir John appears to have been
in doubt whether the greenish yellow colour was to be attributed
to the mixture of the true colour of the ultra-violet rays with the
yellow of the paper due to diffused light, or to the real colour of
the ultra-violet rays themselves, which on that supposition would
have been incorrectly termed “ lavender.”

88 . The fact of the change of refrangibility of light having
been established, there could be little doubt that the true cause of
the extraordinary prolongation of the spectrum on paper washed
with tincture of turmeric was very different from what Sir John
Herschel had supposed, and that it was due to a change of re¬
frangibility in the incident light, which was produced by the
medium in a solid state. Tincture of turmeric has already been
mentioned as a medium which possesses in a high degree the
property of internal dispersion. It was the observation of Sir
John Herschel’s already mentioned, which led me to try this
medium. But it is by no means essential that a sensitive sub¬
stance should be in solution, or in the state of a transparent
solid, in order that the change of refrangibility which it produces
should admit of being established by direct experiment, although
of course the mode of observation must be changed.

89. A piece of paper was prepared by pouring some tincture
of turmeric on it, and allowing it to dry. In this way the part
which was deeply coloured by turmeric was in juxtaposition with
the part which remained white, which was convenient in con¬
trasting the effects of the two portions. The sun’s light being
reflected horizontally into a darkened room through a vertical

* Philosophical Transactions for 1842, p. 194.

21—2

324 ON THE CHANGE OF REFRANGIBILITY OF LIGHT.

slit, the paper was placed in a pure spectrum formed in the
usual manner. On the coloured part the fixed lines were seen
with the utmost facility far beyond the line H, on a yellowish
ground. The colours too of all the more highly refrangible part
of the spectrum were totally changed. From the red end, as far
as the line F, or thereabouts, there was no material change of
colour; but a little further on a very perceptible reddish tinge
came on, which was quite decided at F\G, where it was mixed
with the proper colour of that part of the spectrum. About
6 rfJjT the colour became yellowish. The reality of a change of
refrangibility was easily proved by refracting the spectrum on
the screen by a prism applied to the eye. When the refraction
took place in a plane parallel to the fixed lines, they were seen
distinctly throughout the spectrum ; but when it took place in
a plane perpendicular to the former, the fixed lines in the less
refrangible part of the spectrum, and as far as F, were distinctly
seen; but in the rest of the spectrum they were more or less
confused, or even wholly obliterated, according to their original
strength, the refracting angle and dispersive power of the prism,
and its distance from the paper. With a prism of small angle
the edges of the broad bands H were seen tinged with prismatic
colours.

90. The change of refrangibility was further shown by the
following observation. The paper was placed in the pure spec¬
trum in such a manner that the line of junction of the coloured
and uncoloured parts ran lengthways through the spectrum, so

that the same fixed line was seen partly on the coloured and

partly on the uncoloured portion. On viewing the whole through
a prism of moderate angle applied to the eye, and so held as to
refract the system in a direction perpendicular to tin 1 fixed lines,
the line F was seen uninterrupted, but (r was dislocated, the
portion formed on the yellow part of the paper being a good
deal less refracted than that formed on the white. The latter
was indeed faintly prolonged into the yellow part of the paper,
so that on this part G was seen double; hut the image which

was by far the more intense of the two was less refracted than

that formed on the white paper. The whole appearance was
such as to create a strong suspicion of some illusion, as if some
other group of fixed lines formed on the yellow part of the paper

ON THE CHANGE OF REFRANGIBILITY OF LIGHT.

325

had been mistaken for G , though certainly no reason appears why
such a group should not have had its counterpart on the white
part. However, to remove all doubts, I refracted the system in
the direction of the fixed lines, and then turned the prism round
the axis of the eye through 90°, when the plane of refraction was
situated as before. At first the two portions of the line G were
of course seen in the same straight line; and the perfect con¬
tinuity with which, as the prism turned round, the appearance
changed into what had been first seen, left not the shadow of a
doubt as to the reality of the dislocation.

91. The cause of the whole appearance is plain enough.
The light coming from the illuminated part of the yellow paper
consisted, in the neighbourhood of 6r, of two portions; the first,
indigo light, which had been scattered in the ordinary way; the
second and larger portion, heterogeneous light having a mean
refrangibility a good deal less than that of 6r, which had arisen
from homogeneous light of higher refrangibility. The absence of
the first occasioned the faint prolongation of the more refracted
part of the line G ; the absence of the second gave rise to the less
refracted part.

92. The broad bands H were seen faintly but quite dis¬
tinctly on the white paper. On refracting them sideways by a
prism of moderate angle held to the eye, they became confused,
and tinged with prismatic colours. The confused images of these
bands, seen in the white and coloured parts, were nearly con¬
tinuous. It thus appears that the visibility of the bands H on
the white paper was due to a change of refrangibility which that
substance had produced in violet light of extreme refrangibility.

93. Effects similar to those produced by paper coloured by
tincture of turmeric are also produced by turmeric powder, or
even by the root merely broken across. Notwithstanding the
roughness of the latter, the bands H and fixed lines far beyond
are seen with the utmost facility.

94. These phenomena are much better observed by covering
the slit with a deep blue glass, which absorbs all the bright part
of the spectrum, while it freely transmits the violet and invisible
rays, which are mainly efficient in this class of phenomena. In

326

ON THE CHANGE OF REFRANGIBILITY OF LIGHT.

this way fixed lines may be seen on common white paper far
beyond H. These lines may be seen without the use of the
blue glass, by allowing the bright colours to pass by the edge
of the paper, and receiving on it only the extreme violet and
invisible rays.

95. Paper coloured by turmeric having exhibited so well the
sensibility of that substance, I was induced to try various other
washed papers, in fact, papers washed with most of the fluids
with which I had made experiments. I found almost always
that sensitive solutions gave rise to sensitive papers, exhibiting
a change of refrangibility of the same character as that shown
by the solution. Besides the turmeric paper, the two most re¬
markable were paper washed with a pretty strong solution of
sulphate of quinine, and paper washed with the extract from the
seeds of the Batura stramonium . I should here observe, that it
was not till long after the time when these experiments were
made that I was acquainted with the high sensibility of a decoc¬
tion of the bark of the horse-chestnut. The former of the papers
just mentioned exhibited the fixed lines of the invisible rays on
a blue, and the latter on a green ground. The dispersion pro¬
duced by the quinine paper was not exhibited so early in the
spectrum as in the case of turmeric, nor was it so copious in
the extreme violet rays, and for some distance further on, but
the quinine paper seemed superior to the other for showing the
fixed lines of extreme refrangibility. With the turmeric paper
the group n was plain enough, but with the quinine paper I
have seen some fixed lines of the group p. The stramonium
paper was, on the whole, I think, superior to the quinine paper
in point of the copiousness of the dispersed light, but seemed
hardly equal to it for showing the fixed lines of extreme re¬
frangibility. However, it is likely that paper washed with a
solution of the sensitive principle in a state of purity would
have been quite equal to the quinine paper in this respect.

96. A washed paper is a little more convenient for use than
a solution, but, as might be expected, it does not show the fixed
lines with quite as much delicacy, nor is it quite so good for
tracing the spectrum to the utmost limits to which it can be
traced with the substance employed.

ON THE CHANGE OF REFRANGIBILITY OF LIGHT. 327

97. The sensibility of fresh leaf-green could not be made out
on a washed paper by this mode of observation, but the sensibility
of the substance extracted by alcohol from black tea, from which
the brown colouring matter had been removed by hot water, was
plainly exhibited by the redness which it produced in the highly
refrangible part of the spectrum.

98. Paper washed with a solution of guaiacum seemed an
exception to the general rule ; but this is not to be wondered at,
since a paper prepared in this manner is turned green when
exposed to the light, and it is difficult to prevent some degree
of discoloration. That the fluid state is not essential to the exhi¬
bition of the sensibility of this substance, was however plainly
shown by the high degree of sensibility of the solid resin from
which the solution was made. In this case the bands H were
seen on a greenish ground. The dispersion of a fine blue light
under the influence of rays of still higher refrangibility was
hardly, or not at all, exhibited by the solid resin.

99 . Shell-lac, common resin, glue, are all highly sensitive.
The ground on which the fixed lines in the neighbourhood of H
are seen is brown in the case of shell-lac, and greenish in the case
of resin and glue. The sensibility of glue is evidently not due to
gelatine, for isinglass is almost, if not quite, insensible. These
are merely a few instances of sensibility: I shall defer further
mention of the subject till I have described a better mode of
observation. I will merely observe for the present, that several
washed papers proved not greatly inferior to turmeric paper for
showing the fixed lines about and beyond H.

Effect of refracting a Narrow Spectrum in a Vertical Plane.

100. In the arrangement last described, when a short slit is
used, the spectrum received on the washed paper or other sub¬
stance is of course narrow, so that the fixed lines formed on
the paper are but short, and may roughly be regarded as mere
points. If, now, the whole be viewed through a prism, so as to
be refracted in a vertical plane, the effect is very striking. For
facility of explanation suppose the red to be to the left, and the
rays to be refracted upwards, so that to the observer the image

328

ON THE CHANGE OF REFEANGIBIL1TY OF LIGHT.

is thrown downwards. The original spectrum on the screen is
decomposed by the prism held to the eye into two spectra, which
diverge from each other. The first of these runs obliquely down¬
wards from left to right, and contains the natural colours of the
spectrum from red to violet. It consists of light which has been
scattered in the ordinary way by the substance on which the
primary spectrum is received, and the cause of its obliquity is
evident. The second spectrum is horizontal, that is to say, it
approximates to the form of a long rectangle having its longer
sides horizontal. Of course it would be theoretically possible to
render the vertical sides the longer, but when the whole arrange¬
ment of the apparatus is such as to be convenient for observation,
the horizontal sides are much longer than the others. In this
second spectrum the colours run horizontally , that is to say, the
lines of equal colour are horizontal. The interruptions of the
primary spectrum corresponding to fixed lines, almost reduced
to points, are now elongated, so that in this strangely formed
spectrum the principal fixed lines of the solar spectrum are seen
running across the colours.

101. It will be convenient to have a name for the second of
the two spectra above mentioned. As the term secondary spectrum
is already appropriated to something altogether different, I shall
call it the derived spectrum. The first of the diverging spectra
may be called the primitive spectrum, while the original spectrum,
considered as not yet decomposed by the prism held to the eye,
may be called, for distinction, as in fact it has been already called,
primary.

102 . In accordance with the law enunciated in Art. 80, it is
found that the derived spectrum appears always on one and the
same side of the primitive, being less refracted.

103. The brilliancy of the derived spectrum, its extent, both
vertically and horizontally, the colours of which it mainly consists,
the distribution of its illumination in a horizontal direction, all
depend upon the nature of the substance upon which the primary
spectrum is received. As a general rule, it may be stated that it
starts from the neighbourhood of the brightest part of the primitive
spectrum, and extends from thence onwards to a good distance
beyond the extreme violet; and that with a given substance its

ON THE CHANGE OF REFRANGIBILITY OF LIGHT.

329

colour is pretty uniform, that is, does not much change in passing
from one vertical section to another. Sometimes the derived
spectrum remains very bright np to its junction with the primitive,
or at least till it gets so near that the superior brilliancy of the
primitive spectrum prevents all observation on the derived;
sometimes it remains dull to a considerable distance from the
primitive spectrum, and then, opposite a highly refrangible part
of the primitive spectrum, a strong illumination comes on in the
derived, lasts for some distance, and afterwards gradually dies
away. Many of the results mentioned in this paragraph are
better observed by a somewhat different method, which will
shortly be described.

104. It has been already stated that the bands H were
distinctly seen on common white paper, the substance usually
employed as a screen in experiments on the spectrum, but that
this was due to a change of refrangibility produced in the extreme
violet rays. These same bands have been seen on paper in the
experiments of others, though of course their visibility was not
attributed to its true cause. By the method of observation
described in Art. 100, or still better, by a method not yet explained,
it may be seen that the change of refrangibility produced by
white paper is by no means confined to the extreme violet rays,
and those still more refrangible, but extends from about the
middle of the spectrum to a good distance beyond the extreme
violet. The distance to which the illumination can be traced by
means of light merely scattered in the ordinary way, may be seen
by examining the primitive spectrum. In the primitive spectrum
formed on white paper and other white substances, I have not
been able to trace the illumination beyond the edge of the broad
band H , which accords very well with the illuminating power of
the extreme violet when received directly into the eye.

Illuminating Power of the Rays of High Refrangibility .

105. The prolongation of the spectrum seen on turmeric
paper was brought forward by Sir John Herschel as a proof of the
visibility of the ultra-violet rays, or rather as a confirmation of
other experiments which had led him to the same conclusion.
Of course, the experiment with turmeric must now be regarded as

330

ON THE CHANGE OF REFRANGIBILITY OF LIGHT.

having no bearing on the question; but from the way in which
Sir John speaks of it, it would appear that he thought the other
experiments not so conclusive as to be independent of the con¬
firmation which they received from this. The experiment with
the distorted spectrum, indeed, must now be put out of account,
because in this experiment, as I have been informed by Sir John
Herschel, the light was only thrown on a screen. Accordingly,
the question of the visibility of these rays may be regarded as
open to further investigation.

While engaged in some of the experiments described in Art. 89,
I had occasion to form a pure spectrum in air in a well-darkened
room, the slit itself by which the suns rays entered being covered
by a deep blue glass, so that no great quantity of light entered
even at this quarter. Now, if ever, it would appear that the
ultra-violet rays ought to be seen by receiving them directly into
the eye; for the blue glass was so transparent with regard to these
rays that the fixed lines far beyond H were seen with facility,
even on substances, such as white paper, which stand low in the
scale of sensibility; and the length of the spectrum from B to H
was about an inch and a quarter, so that when the extreme violet
rays entered the pupil, supposed to be held near the pure spectrum,
not only the extreme red rays transmitted by the blue glass, but
even the brighter part of the transmitted blue and violet rays, fell
altogether outside it. However, on holding the eye a few inches
in front of the pun; spectrum, so as to see the fixed lines distinctly,
the bands 11 were indeed seen with great, facility; hut I was not
able to make; out fixed lines beyond the end of t.lie group l } that
is, about the end of Fraunhofer’s map. However, tin* (yes of
different individuals may differ much in their power of being
affected by tin 1 highly refrangible, rays. It must be confessed
that, on looking in tin* direction of the prisms, a, good deal of blue
light was seen, consisting of light, which had been scattered at the
surfaces of tin 1 prisms and lens. This light, f hough far from
dazzling, was sufficient, to prevent the eye from seeing excessively
faint objects, even flmugh they might be well defined. For
want of a heliosfat, I did not attempt, an experiment I was
meditating for securing a more perfect isolation of the ultra-violet
rays*.

H See note B.

ON THE CHANGE OF REFRANGIBILITY OF LIGHT. 331

However, it seems to me to be a point of small importance, so
far as regards its bearing on other physical questions, whether the
illuminating power of these rays is absolutely null or only ex¬
cessively feeble. It is quite certain that, if not absolutely null,
their illuminating power is at least utterly disproportionate to the
effect which they produce in the phenomena to which the present
paper relates, and indeed that is true even of the violet rays. By
illuminating power, I mean of course power of producing the
sensation of light when received directly into the eye; for by
giving rise to light of lower refrangibility, they are able to
illuminate strongly an object on which they fall.

Mode of Observation specially applicable to Opaque Bodies.

106. In some of the experiments already described, the
change of refrangibility was exhibited, which was produced by
washed papers and solid bodies. There exists, however, a mode
of observation far preferable to those which have already been
explained as applicable to such cases, and which may even in
some instances be employed with advantage in the examination
of transparent bodies. In the experiment described in Art. 100,
the primitive spectrum is pure, but the derived spectrum impure,
on account of the finite length of the slit. Were the slit reduced
to a point, it is true that the derived spectrum would become pure
like the primitive, but then the quantity of light would be so
small that the primary spectrum would hardly bear prismatic
analysis. It is well, once for all, to examine a few sensitive opaque
substances in a very pure spectrum, because then the exhibition
of fixed lines running across the colours in the derived spectrum
removes even the shadow of a doubt as to the reality of the change
of refrangibility of the incident light. Besides this, the only
theoretical advantage in having the primitive spectrum very pure
is, that it might be expected to enable us to detect any very rapid
fluctuations in the colour or intensity of the dispersed light. Of
course, I am now speaking only with reference to experiments in
which the observer is employing the spectrum to examine some
substance, not employing the substance to examine the spectrum.
But practically, I have not found any advantage on this account;
for abrupt, or almost abrupt, changes in the colour or intensity of
the dispersed light hardly ever, if ever, occur, except when the

332

ON THE CHANGE OF REFRANGIBILITY OF LIGHT.

active and the dispersed light have very nearly the same refrangi-
bility. But such changes could not be observed even with a pure
primitive spectrum, because in the place where they occur the
primitive and derived spectra overlap; and independently of this,
the brilliancy of the primitive spectrum would prevent all exact
observation of the derived. It is true that, in the case of chloro¬
phyll, or some of its modifications, changes of intensity having
apparently somewhat the same nature were observed when the
active and the dispersed light were widely separated in refrangi-
bility. But the sensibility of this substance is difficult, if not
impossible, to observe in the case of a washed paper or a green
leaf, except by one of the methods not yet described, so that it is
not to be expected that such fluctuations could be made out.
Besides, it is to be remembered that the fluctuations observed in
the case of solutions of chlorophyll were fluctuations in the rate
at which dispersed light was produced, not fluctuations in the sum
total of the dispersed light produced by the time the active light
was exhausted. Fluctuations of the former kind by no means
imply fluctuations of the latter ; and indeed the (‘ironinstance, that
maxima of activity in the solution correspond to minima of trans¬
parency, would seem to show that the total quantity of light
dispersed, considered as a function of the refrangibility of the
active light, is not subject to these fluctuations, or at, least not to
anything like the same extent. Now the total quantify of red
light dispersed by a green half, or by a paper washed with a
solution of chlorophyll, must depend upon tin*, sensibility of this
substance and upon its transparency conjointly, and therefore it is
likely enough that such maxima, and minima would not. lx* observed,
even were the dispersed light much stronger than it. is.

107. Suppose now the slit by which tin* light enters to be
placed in a horizontal instead of a vertical position, so as t.o lie in
the plane of refraction. Corresponding to light of any given
refrangibility, the image of the slit formed after refraction through
the prisms and lens will now bo a narrow parallelogram, which may
be regarded as a horizontal lino. The series of these lines, suc¬
ceeding one another in a horizontal direction, and consequently
overlapping, forms the spectrum incident on the body examined.
This spectrum is now no longin' pure, hut only approximately so,
a point, however, which, as we have seen, is not. of much con-

ON THE CHANGE OF REFRANGIBILITY OF LIGHT.

333

sequence. But by this trifling sacrifice two very great advantages
are gained. The first is increase of illumination. When the slit
is vertical, the spectrum received on the body occupies a rectangle
having for breadth the length of the image of the slit; but when
it is horizontal, the same, or very nearly the same, quantity of
light is concentrated into a rectangle having the same length as
before (the length of the image of the slit being disregarded com¬
pared with that of the spectrum), but having for its breadth only
the length of the image of a line drawn across the slit. Hence
the intensity of the incident light is increased in the ratio of the
breadth to the length of the slit. The second advantage is purity
in the derived spectrum, a point of much consequence, because
sometimes the composition of this spectrum presents very remark¬
able peculiarities. If the slit be not too long, the spectrum
formed in air is still sufficiently pure to allow us to make out
in a general way what are the refrangibilities of those portions
of the incident light which are most efficient in producing dis¬
persed light; and this is nearly all that can be done even when
the spectrum is very pure.

108. The method of observation which has just been de¬
scribed is that which latterly I have almost exclusively employed
in examining opaque substances. As it will be convenient to have
a name for it, I shall speak of examining a substance in a linear
spectrum. In examining substances which are only slightly
sensitive, it is often highly advantageous to cover the slit with
a blue glass.

109. Fig. 5 is intended to represent the usual appearance
of the primary linear spectrum, and of the primitive and derived
spectra. IT is the primary spectrum, as seen by the naked
eye, RV t ST are the primitive and derived spectra into which it
is separated by the prism held to the eye. The direction of the
shading in RV is intended to represent the composition of this
spectrum, which may be regarded as consisting of an infinite
number of images of the slit arranged obliquely in the order of
their refrangibility. The direction of the shading in ST is that
of the lines of the same colour and same refrangibility. Of course
the figure does not represent the amount of vertical displacement
of the primary spectrum when viewed through the prism held to
the eye.

334 ON THE CHANGE OP REFRANGIBILITY OF LIGHT.

110 . There is another mode of observation which I have
occasionally found convenient when the object was to determine
whether a substance exhibited so much as a low degree of
sensibility. In this method the sun's light was reflected hori¬
zontally through a large lens, and then transmitted through a
small lens placed in the condensed beam. The small lens was
covered by a small vessel with parallel sides of glass, containing
a blue ammoniacal solution of copper, or else by a deep blue
glass combined with a weak solution of nitrate or sulphate of
copper. The object of the latter solution was to absorb the
extreme red which is transmitted by a blue glass. The light
coming through the lens was then analysed by a prism, being
received directly into the eye, or else allowed to fall on a white
object which had been previously ascertained not to change the
refrangibility of the light incident upon it. I found clean white
earthenware to serve very well for such an object, but each
observer ought to test for himself the substance he employs.
When a test object, such as white earthenware, is used, it is
placed at the focus of the lens, and the spot of blue light formed
upon it is analysed by a prism to see if the absorption is
sufficient. When the visible rays are considered to have been
sufficiently absorbed, the object to be observed is placed at the
focus of the lens, and the spot of light formed upon it is viewed
through a prism. The spectrum then seen is compared with that
given by the test object. This method of observation is rather
easier than that of a linear spectrum, and is at least as delicate
if the object be merely to determine whether a substance is
sensitive or not, but on the whole it is not near so useful. It
may sometimes be used with advantage in the case of translucent
bodies.

111 . An extremely pale solution of nitrate or sulphate of
copper is sufficient to absorb the extreme red transmitted by a
deep blue glass. This is not the case with the ammoniacal
solution, which does not absorb the extreme red till it is of a
pretty deep blue. Its absorbing power is greatest, not at the
extreme red, but about the orange, as may be seen by using
candle-light, which is richer in red rays than daylight.

112 . Another method of observation which is sometimes
useful, consists in employing a large lens and absorbing medium,

ON THE CHANGE OF RE FRAN GJBILIT Y OF LIGHT.

335

as described in Art. 110, but leaving out the additional small
lens. The substance to be examined is placed in the condensed
beam, and viewed through an absorbing medium which is ap¬
proximately complementary to the former. This method is chiefly
useful in examining a confused mass of various substances. The
most minute fragments of sensitive substances show themselves in
this manner.

Results obtained with a Linear Spectrum .

113. When this method is applied to the examination of
common objects, it is found that the property of producing a
change of refrangibility in the incident light is extremely common.
Thus, wood of various kinds, cork, horn, bone, ivory, white shells,
leather, quills, white feathers, white bristles, the skin of the hand,
the nails, are all more or less sensitive. To make a list of
sensitive substances would be endless work; for it is very rare
to meet with a white or light-coloured organic substance which
is not more or less sensitive. I am not now speaking of organic
substances obtained in a state of chemical isolation, of which
some are sensitive and others insensible. That substances of a
dark colour should frequently prove insensible is only what
might have been expected, because the dispersed light is not
reflected from the surface, but emanates from all points of a
stratum of finite thickness; and in order that dispersed light
should be forthcoming, it is necessary that the active light
entering, and the dispersed light of a different refrangibility
returning, should both escape absorption on the part of the
colouring matter. Such substances usually consist of a mixture
of various chemical ingredients, of which one or more may very
likely be sensitive, in which case the substance may be compared
to a solution of sulphate of quinine mixed with ink. Frequently
however the colouring matter is itself sensitive.

114. Among sensitive substances I have mentioned the skin
of the hand, which stands rather low in the scale. I have found
the back of the hand a convenient test object. When the sun¬
light is not strong enough to show with ease the derived spectrum
in the case of the hand, there is little use in attempting to
observe.

336 ON THE CHANGE OF REFRANGIBILITY OF LIGHT.

115. It is needless to say that papers washed with tincture
of turmeric, or with a solution of sulphate of quinine, display
their sensibility in a remarkable manner when examined in a
linear spectrum. The sensibility of turmeric paper is rather
impaired by exposing the paper to the light, but on the other
hand is materially increased by washing it with a solution of
tartaric acid.

116. Paper washed with an ethereal solution from dried
archil exhibited very well ""the sensibility of that substance. The
derived spectrum consisted chiefly of two distinct portions, one
containing orange and a little red, the other consisting chiefly
of green, just as in the beam of dispersed light, produced by
white light taken as a whole, which the solution itself exhibited.
Indeed, I have found that the prismatic composition of dispersed
light could be determined even more conveniently by means of a
linear spectrum than by means of the beam dispersed by a
solution.

117. The inside of the capsules of the Datura stramonium
is nearly white, and apparently uniform. But when the capsules
are examined in a linear spectrum, certain patches shine out
like bright clouds in the invisible rays. The whole of the inside
is sensitive, as such substances almost always are, but these
patches, which are probably spots against which the seeds have
pressed, are remarkably so. The capsules were examined after
they had begun to burst.

118. By means of a linear spectrum the sensibility of chloro¬
phyll may be detected in a green leaf. It is exhibited by the
appearance in the derived spectrum of a narrow pure red band
of remarkably low refrangibility. The refrangibility is so low
that I have always found this band separated from the derived
spectrum due to other sensitive substances with which chlorophyll
or one of its modifications might have been mixed.

119. The petals of flowers, so far as I have examined, are as
a class rather remarkable for their insensibility, some appearing
quite insensible, and others only slightly sensitive. The bright
yellow chaffy involucre of a species of everlasting, proved, how¬
ever, highly sensitive, and its sensibility was also displayed in

ON THE CHANGE OF REFKANG1BILITY OF LIGHT. 337

an alcoholic solution. This medium was sensitive enough to
exhibit a pretty copious dispersive reflexion of a pale greenish
yellow light. Its sensibility was more confined than usual to the
rays of very high refrangibility.

120 . Among petals, the most remarkable which I have
observed are those of the purple groundsel (Senecio elegans ).
These petals disperse a red light, more copious than is usual
among petals. If a petal be placed behind a slit, and the
transmitted light be analysed, it is 'found to exhibit three re¬
markable bands of absorption, much resembling those of blue
glass, but closer together, and beginning later in the spectrum,
the first appearing about the place of the orange. These bands
are still better seen in a solution of the colouring matter in weak
alcohol. On examining this medium by the third method, with
a lens of shorter focus than usual, and looking down from above,
the places of the absorption bands were indicated by tooth-shaped
interruptions in the beam of light reflected from motes. The
points of these teeth were occupied by red dispersed light, which
did not appear in the intervening beams of light reflected from
motes, from whence it appears that there is the same sort of
connexion between the absorption and dispersion of this medium
as was noticed in Art. 59, in the case of solutions of chlorophyll
and its modifications.

121. A collection of sea-weeds appeared all more or less
sensitive, most of them highly so. All, or almost all, except the
white ones, exhibited in the derived spectrum the peculiar red
band indicative of chlorophyll and its modifications. The trans¬
mitted light also exhibited more or less the absorption bands
due to this substance, which was likewise, in the specimens tried,
extracted by alcohol. But the most remarkable example of
sensibility found in sea-weeds occurs in the case of the red
colouring matter contained in orangy red, red, pink, and purple
sea-weeds. To judge by its optical properties, this colouring
matter appears to be the same in all cases, but to be mixed in
different proportions with chlorophyll, or some modification of it,
and probably other colouring matters, thus giving rise to the
various tints seen in such sea-weeds. The derived spectrum
exhibited by sea-weeds of this kind consists mainly of a band
of unusual brightness, containing some red, followed by orange

22

s. III.

338

ON THE CHANGE OF REFRANGIBILITY OF LIGHT.

and yellow. This band fades away gradually at its less re¬
frangible limit, where it is separated by a dark interval from the
narrow well-defined red band of still lower refrangibility due to
chlorophyll. At its more refrangible limit, however, it breaks
off with unusual abruptness.

122 . When the light transmitted through such a sea-weed
is subjected to prismatic analysis, in addition to one at least of
the absorption bands due to chlorophyll, there is seen a band
obliterating the yellow, another dividing the green from the blue,
and a third, far less conspicuous, dividing the green into two.
The whole of the green is absorbed more rapidly than the blue
beyond, and not merely than the red, which last is the final tint.

123. The red colouring matter is easily extracted by cold
water from certain kinds of red sea-weed, if fresh gathered; but
when once the plant has been.dried, the colouring matter cannot
be extracted in any way that I know of. It is apparently
insoluble in alcohol and ether, and is decomposed by boiling.
Cold water extracts only a trace of it after a long time.

124. A piece of recently gathered red sea-weed, on being
mashed with cold water, readily gave out its red colouring
matter. When the residue was treated with alcohol, the fluid
was almost immediately coloured green by chlorophyll, whereas
this substance is only very slowly and sparingly extracted by
alcohol from dried sea-weeds. A dried sea-weed may apparently
be assimilated to an intimate mixture of gum and resin, which
it would be very difficult to dissolve, whether it were attacked by
water or alcohol.

125. The solution of the red colouring matter was highly
sensitive, exhibiting a copious dispersive reflexion of a yellowish
orange light. The transmitted light was pink or red, according
to the thickness through which the light passed. When this
light was analysed, the same three absorption bands which have
been already mentioned were perceived. The analysis of the
light transmitted by the fronds of various red sea-weeds had
rendered it extremely probable that the faint division in the
green did belong to the red colouring matter; but till I had
obtained this matter in solution I did not feel certain that it

ON THE CHANGE OF KEFRANGIBILITY OF LIGHT.

339

might not have been due to chlorophyll, the spectrum of which
exhibits a division in the green.

126. When this fluid was examined in Sir David Brewsters
manner, and the dispersed beam was analysed, the spectrum was
found to consist of a broad band like that which has been already
described as seen in the derived spectrum given by a frond of red
sea-weed. When the solution, which happened to be very weak,
was examined by the third method, the dispersion was found to
be produced chiefly by a portion of the incident spectrum, having
a breadth about equal to that of the interval between the two
principal bands of absorption. To each of these bands cor¬
responded a maximum of activity. The tint of the dispersed light
was nearly uniform; but by the fourth method of observation
some faint dispersed red could be made out, which appeared
before the main part of the dispersion had come on. This
medium affords a very good example of an intimate connexion
between absorption and internal dispersion.

127. The colouring matters of bird's feathers appeared to be
insensible, white feathers being most sensitive, pale ones next,
and dark ones not at all: however, I have not examined a large
collection.

128. Of coloured fruits, such as currants, &c., the colouring
matter appeared, in the very few cases which I have examined, to
be quite insensible.

129. A set of water colours were by no means remarkable for
sensibility, but rather the contrary. The inorganic colours
appeared quite insensible, except white lead, the sensibility of
which was perhaps due to size*, and offered nothing striking,
either as to its character or as to its amount. Some lakes and
other organic colours proved moderately sensitive. But I found
one water colour, called Indian yellow, which stands pretty high
among sensitive substances. In its mode of dispersion it much
resembles turmeric, but it does not come up to that substance in
the amount of sensibility. It is said to be composed of urate of

[* Meaning, of course, not magnitude, but the substance used to make the
powder stick together in a cake.]

22—2

340

ON THE CHANGE OF REFRANGIBIUTY OF LIGHT.

lime*, but I do not know how far it may be regarded as chemically
pure.

130. Many of the substances used in dyeing, and dyed articles
in common use, furnish very remarkable examples of sensibility.
Archil, litmus and turmeric have been already mentioned; and I
have been recently informed by a friend that the Mercu/rialis
perennis, in which a striking instance of sensibility was observed,
was formerly employed in dyeing. A piece of scarlet cloth,
examined in a linear .spectrum, gave a copious derived spectrum
which was very narrow, consisting chiefly of the more refrangible
red. With a vertical slit the bands H and fixed lines beyond
were seen on a red ground. Paper washed with a solution of
cochineal and afterwards with a solution of alum, when examined
in a linear spectrum, displayed a pretty high degree of sensibility,
the derived spectrum consisting in this case of a red band. If
tartaric acid be used instead of alum, the dispersion is a good deal
more copious.

Common red tape is another example in which the derived
spectrum is very copious, consisting mainly of a red band. Some
red wool, dyed I suppose with madder, proved extremely sensitive.
The derived spectrum in this case was pretty broad, but red was
the predominant colour. Green wool, dyed I do not know with
what, was also very sensitive, giving a pretty broad derived
spectrum, in which green was the predominant colour. These
examples may suffice, but the reader must not suppose that they
form the only instances in which dispersion was observed among
dyed substances. On the contrary, it is extremely common in this
class.

131. Brazil wood, safflower, red sandal wood, fustic and
madder, all gave rise to solutions having a pretty high degree
of sensibility. The solutions here referred to were such as were
obtained directly by water, &c., in which the colours which these
substances are capable of producing were not brought out. The
beautiful red colouring matters of logwood and camwood appear
to be insensible; for a fresh-made solution of logwood in water
exhibited no perceptible sensibility, and the slight sensibility

[* l^ 3 was 80 stated in a book on artists’ colours, but Dr Stenhouse told me it
was a lake of some kind.]

ON THE CHANGE OF REFRANGIBILITY OF LIGHT.

341

exhibited by a similar solution of camwood seemed to have no
relation to the red colouring matter.

132. Paper washed with a solution of madder in alcohol
was sensitive in a pretty high degree, but the sensibility was
greatly increased by afterwards washing with a solution of alum.
Accordingly I found that a decoction of madder in a solution of
alum exhibited a very high degree of sensibility, displaying a
copious dispersive reflexion of a yellow light. In this medium the
dispersion commenced about the fixed line D, and continued from
thence onwards far beyond the extreme violet, so that the group
of fixed lines n was seen with great ease.

133. Safflower red, examined in the shape in which it is sold
on what is called a pink saucer , proved highly sensitive, giving a
bright and narrow derived spectrum, which consisted chiefly of
the more refrangible red. This substance possesses some other
remarkable optical properties, which however do not belong to
the immediate subject of this paper.

134. Metals proved totally insensible. I have examined gold,
platinum, silver, mercury, copper, iron, lead, zinc and tin. Brass
is like simple metals in this respect; but if the surface be lackered
the lacker displays its own sensibility.

135. The non-metallic elements, carbon, sulphur, iodine and
bromine, are insensible.

136. Among common stones I have found dark flint, lime¬
stone, chalk and some others which were sensitive, though only in
a low degree compared with organic substances. To guard against
any impurity of the surface, the stones were broken across, and
the fresh surface examined. In the cases mentioned the sensibility
observed is not to be attributed to the chief ingredient of the
stone, for quartz, chalcedony, Iceland spar and Carrara marble
were insensible.

Compounds of Uranium.

137. Towards the end of last autumn, when the lateness of
the season afforded but few opportunities for observation, I learned
from different sources that the kind of yellow glass which has
been already mentioned as possessing in so high a degree the

342

ON THE CHANGE OF RE FRANG IJBILIT Y OF LIGHT.

property of internal dispersion was coloured with oxide of
uranium. This rendered it interesting to examine other com¬
pounds of uranium; and I accordingly procured some crystallized
nitrate of the peroxide, which, with a few other compounds formed
from it, and some of the natural minerals which contain uranium,
were examined by methods which have been already explained.

138. The crystals of the nitrate were not sufficiently large
and perfect to admit of observation by the methods applicable to
fluids and clear solids, but they could be readily observed by
means of a linear spectrum. They proved to be sensitive in a
very high degree, dispersing a green light which had the same
very remarkable composition that has been already described in
the case of the yellow glass. On placing a crystal in the continu¬
ation of the same linear spectrum with the glass, and viewing the
whole through a prism, the five bright bands of which the derived
spectrum given by each of the two media usually consisted,
appeared to correspond to one another as regards their position in
the spectrum. With great concentration of light J have seen an
additional band of greater refrangibility in the spectrum of the
crystals.

139. Some crystals of nitrate of uranium were gently heated
so as to expel a good part at least of the water of crystallization.
The residue after some time became opaque and nearly white. In
this state it was still more sensitive than the crystals. The
dispersed light was not exactly of the same tint, but more nearly
white; and the derived spectrum was found on being analysed to
contain, in addition to the bright bands usually soon in the
derived spectrum of the crystals, another blue hand still more
refrangible. The fused mass gradually attracted moisture from
the air, its colour changed to that of the crystals, and the most
refrangible of the bright bands disappeared from the derived
spectrum. Although when the incident light was very much
concentrated I have seen this band even in the crystals, it was
faint compared with the preceding bands, whereas in the ease
of the whitish mass its intensity was not very different from
that of the others. It appears therefore that the quality as well
as the quantity of the dispersed light was altered by depriving the
crystals of a part of their water.

ON THE CHANGE OF REFRANGIBILITY OF LIGHT.

343

140. A solution of nitrate of uranium in water is decidedly
sensitive, though not sufficiently so to exhibit much dispersive
reflexion. When the dispersed beam is analysed it is resolved
into bright bands. When the solution is examined in a pure
spectrum, the mode of dispersion is found to agree with that
of canary glass. The dispersion commences abruptly at the same
part of the spectrum as in the case of the glass, and after a rather
narrow band in which light is copiously dispersed, there follows a
remarkable minimum of sensibility, just as in the glass (see Art.
76), where the dispersed light is almost imperceptible. After this
the dispersion is resumed, and offers nothing remarkable. The
minimum of sensibility occurs at the very same place in the
spectrum, whether the sensitive medium be a solution of nitrate
of uranium or glass coloured yellow by uranium.

141. Yellow Uranite .—This mineral, when examined in a
linear spectrum, proved to be sensitive in an extremely high
degree. The derived spectrum consisted, as in the case of the
glass, of bright bands arranged at regular intervals, but in this
case six were seen, a band being visible in the faint red at the
extremity of the spectrum which could not be made out in the
case of the glass.

142. Green Uranite , or Chalcolite .—According to M. Peligot
the formula of the yellow uranite of Autun is PhO 5 , CaO, 2(U 2 0 2 0),
8HO, and the green uranite differs from the yellow only in having
the lime replaced by oxide of copper*. Yet a specimen of green
uranite on being examined in a linear spectrum proved totally
insensible. The primitive spectrum showed however a very
remarkable system of dark bands depending on the absorption
of light by the mineral. In examining these bands, the previous
prismatic decomposition of the light, so far from being necessary,
is decidedly inconvenient. It is better to dispense with the
prisms altogether, using only the lens, and placing the mineral
so that the image of the slit is formed upon it. The bright line
thus formed is viewed from a convenient distance through a
prism, the eye being held out of the direction of regular reflexion.
The position of any bands which may appear in the spectrum can
then be determined by means of the fixed lines, which are seen at

* Annales da Chimie, Tom. v. (1842), p. 46.

344

ON THE CHANGE OF REFRANGtBILITY OF LIGHT.

the same time; or, if it be desired to see the latter more dis¬
tinctly, it will be sufficient to attach a fragment of paper to the
mineral or other substance, placing it so that the image of the slit
is formed partly on the paper and partly on the substance to be
examined. I have frequently found this mode of observation
convenient in examining the absorption of light by opaque
substances. The manner in which the absorption of the medium
comes into play in this case will be considered in greater detail
further on (see Art. 176).

143. When green uranite was examined in this manner, it
showed a very remarkable system of dark bands of absorption.
These bands were seven in number, or at any rate six, and were
arranged with all the regularity of bands of interference. The first
was situated at about bfF y the second at F; the middle of the
sixth fell a very little short of G ; the third, fourth and fifth were
arranged at regular intervals between the 1 second and sixth; the
seventh was situated about as far beyond the sixth as the sixth
beyond the fifth. The spectrum was so faint in the region of the
seventh band as to leave some slight doubts respecting its exist¬
ence. There would not have been light enough to see bands
further on.

144. Uranite is highly lamellar in its structure, from whence
it is otherwise called uran-mica. The reader may perhaps suppose
that the dark bands described in the last paragraph were, bands of
interference, which I had mistaken for bands of absorption, and
that they were really of the nature, of Newtons rings, or more
exactly of the bands .seen in an experiment due to the Baron von
Wrede. There may, it will perhaps be said, have been a fissure
parallel to the first surface, so as to separate a, thin plate ; and the
interference of the two streams of light reflected respectively on
the upper and under surface of this plate may have produced the
bands observed. But various phenomena attending these bands
are irreconcilable with such a supposition. Towards tin* edges of
the crystal, where flaws did in fact exist, bands of the same nature
as Von Wrede s were actually observed. But these had an
appearance totally different from that of the others. The dark
bands of the interference system were more intensely black and
better defined than those of the other system, and were very

ON THE CHANGE OF REFRANGIBILITY OF LIGHT.

345

variable, depending as they did upon the thickness of the plate
by which they were formed, whereas the bands belonging to the
first system were always the same. Besides, were these bands due
to interference, there is no reason why they should be confined to
one region of the spectrum, and that by no means the brightest.
However, to take away all possible doubts respecting the nature
of the bands, I detached a small scale from the crystal, and having
placed it behind a slit in a beam of sunlight condensed by a lens,

I analysed the transmitted light by a prism. Were the bands
really due to absorption, they ought to be more distinct in the
transmitted light, whereas, were they of the nature of Yon Wrede’s
bands, they ought to be faint, and almost imperceptible. The
spectrum of the transmitted light contained however four dark
bands, which were well defined and intensely black. The whole
of the spectrum beyond the place of the next band was absorbed,
which is the reason why four bands only were visible.

145. The absorption bands of green uranite, though they
showed great regularity with respect to their positions, did not
appear very regular with regard to their intensities. The second,
fifth and sixth seemed to me to be more conspicuous than the
first, third and fourth. I cannot say for certain whether this
ought to be attributed to fluctuations in the absorbing power of
the medium, or fluctuations in the original intensity of the solar
spectrum, but I am strongly inclined to prefer the former view.

146. The intervals between the absorption bands of green
uranite were nearly equal to the intervals between the bright
bands of which the derived spectrum consisted in the case of
yellow uranite. After having seen both systems, I could not fail
to be impressed with the conviction of a most intimate connexion
between the causes of the two phenomena, unconnected as at first
sight they might appear. The more I examined the compounds
of uranium, the more this conviction was strengthened in my
mind.

147. Yellow uranite exhibits a system of absorption bands
similar to those of green uranite. Nitrate of uranium also shows a
similar system. In a solution I have observed seven of these bands
arranged at regular intervals. The first absorption band coincided
with F, the fifth with G nearly. The absorption bands may also

346 ON THE CHANGE OF REFRANGIBILITY OF LIGHT.

be seen by analysing the light transmitted through the crystals.
The following arrangement exhibited at one view the absorption
bands and those due to the light which had changed its re-
frangibility.

148. The sun’s light was reflected horizontally by a mirror,
and condensed by passing through a large lens. It was then
transmitted through a vessel with parallel sides containing a
moderately strong ammoniacal solution of a salt of copper. The
strength of the solution, and the length of the path of the light
within it, were such as to allow of the transmission of a little
green besides the blue and violet. A crystal of nitrate of uranium
was then attached to a narrow slit, and placed in the blue beam
which had been transmitted through the solution, the crystal
being turned towards the incident light. The light coming from
the crystal through the slit was then viewed from behind, and
analysed by a prism. A most remarkable spectrum was thus
exhibited, consisting from end to end of nothing but bands
arranged at regular intervals. The interval between consecutive
bands appeared to increase gradually from the red to the violet,
just as is the case with bands of interference. Although this
interval appeared to alter continuously from one end of the
spectrum to the other, the entire system of bands was made up
of two distinct systems, different in appearance, and very different
in nature. The less refrangible part of the spectrum, where only
for the crystal there would have been nothing but darkness, was
filled with narrow bright bands, due to the light which had
changed its refrangibility. These bands were much narrower
than the dark intervals between them, but they were not mere
lines containing light of definite refrangibility. The more re¬
frangible part of the spectrum was occupied by the system of
bands of absorption. The interval between the most refrangible
bright band and the least refrangible dark band of absorption
appeared to be a very little greater than one band-interval, so
that had there been one band more of either kind the least
refrangible absorption band would have been situated immediately
above the most refrangible bright band. With strong light I
think I have seen an additional band of this nature.

149. Pitchblende .—This mineral proved to be quite insensible,
and exhibited nothing remarkable.

ON THE CHANGE OF HE FEANGIBILIT Y OF LIGHT.

347

150. Hydrate of Peroxide of Uranium. —Some crystallized
nitrate of uranium was exposed to a heat a good deal short of
redness, whereby most of the acid was expelled. The residue was
of a deep brick-red colour, and consisted no doubt chiefly of
anhydrous peroxide. It was quite insensible. In order to remove
any undecomposed nitrate, it was boiled with water, whereby the
undecomposed nitrate was dissolved, and the peroxide converted
into a hydrate. This hydrate, after having been washed and
dried at the temperature of the air, was of an extremely beautiful
yellow colour, and was I suppose the hydrate IPO 3 4- 2HO
described in chemical treatises. It was tolerably sensitive, in fact
for an inorganic substance extremely so, though the sensibility
was much less than that of nitrate of uranium, yellow uranite, or
canary glass. The derived spectrum consisted as before of separate
bright bands. A small portion of the powder was attached by
water to blotting-paper, and dried before a fire. The powder thus
obtained on paper was duller than before, and inclined a little
more to orange, though the colour was not much deeper than that
of the former hydrate. From its colour and the circumstances of
its formation, it was probably the other hydrate U 2 0 3 -fH0. It
proved on examination to be totally insensible.

151. Acetate of Peroxide of Uranium , prepared by dissolving
the yellow hydrate of the peroxide in acetic acid, and evaporating
to crystallize.—This salt is extremely sensitive, about as much so
as the nitrate. The derived spectrum consisted of six bright
bands arranged at regular intervals. It seemed to me that the
last five of these were respectively a little more refrangible than
the five bands given by the nitrate, and then a sixth band was
visible in the faint red in the case of the acetate which was not
ordinarily seen in the nitrate. However, this observation has
need to be repeated under more favourable circumstances.

152. Nitrate and acetate of peroxide of uranium, yellow
uranite, and canary glass, are all so highly sensitive as to allow
the primary spectrum to be examined with a prism at some
distance. In the first three media the bright bands are narrow,
much narrower than the dark intervals between; in the glass
they appear much broader than in the other media.

153. Oxalate of Peroxide of Uranium , prepared in the manner
mentioned by M. Peligot, namely, by adding a saturated solution

348

ON THE CHANGE OF REFEANGIBILITY OF LIGHT.

of oxalic acid to a solution of nitrate of uranium, washing and
drying the precipitate.—This salt was sensitive, but only in a low
degree. However, the derived spectrum bore prismatic examina¬
tion sufficiently to show three or four bright bands. The ab¬
sorption of the medium was examined by spreading some of the
powder on glass along with water and allowing it to dry. The
layer was then examined by different methods. The salt exhibits
three very intense absorption bands in the highly refrangible part
of the spectrum. The positions of these bands, by measurement,
were .FO'31 G, F 0*58 G, F 0*85 G.

154 Phosphate of Peroxide of Uranium, prepared by pre¬
cipitation from a solution of nitrate of uranium by adding a
solution of common phosphate of soda.—This salt was sensitive,
though not in a high degree. It was a good deal more sensitive
than the oxalate, but I think not so much so as the hydrate of
the peroxide. The derived spectrum consisted of bright bands as
usual *.

155. Uranate of Potassa, prepared by dropping a solution of
nitrate of uranium into a solution of caustic potash, stopping long
before the alkali was neutralized.—This salt was found to be
insensible, both in its original state as a gelatinous hydrate, and
in various stages of drying.

156. Uranate of Lime, prepared in a similar manner with
lime-water.—This salt, which after drying is of a line orange
colour, was like the preceding found to be insensible. It seemed
interesting to examine these two salts, because tint former contains
two elements (not counting oxygen) in common with canary glass,
and the latter two elements in common with yellow u rani to. Yet
the salts are insensible while the two other media, are so remark¬
ably sensitive.

157. Solutions by means of Alkaline Carbonates .— It is known
to chemists that alkaline carbonates, added in solution to a
solution of nitrate of uranium, give yellow precipitates which are
redissolved in an excess of the precipitant. The solutions thus
obtained with the carbonates of potassa and soda, which were of
a greenish yellow colour, were found to be totally insensible.

* See note C.

ON THE CHANGE OF RE FRAN GIBILITY OF LIGHT.

349

They exhibited however four of those singular absorption bands
so characteristic of salts of peroxide of uranium. Of these the
third fell a little short of G, its more refrangible edge nearly
coinciding with that fixed line; the first and second were situated
between F and G, the distance of the first beyond F being some¬
what greater than the interval between two consecutive bands.
The fourth, which was situated beyond G, was fainter than the
others. The second and third were the most conspicuous of the
set.

158. The absorption bands due to peroxide of uranium afford
an easy mode of detecting that substance in solution. For this
purpose the solutions mentioned in the preceding paragraph are
much preferable to the nitrate, for they produce much stronger
bands when only a small quantity of uranium is present. The
absorption bands of nitrate of uranium are visible, as might
have been expected, in presence of a large quantity of nitrate of
copper*.

Optical Tests of Uranium in Blow-pipe Experiments .

159. When a bead of microcosmic salt is fused with oxide of
uranium, and brought to its highest state of oxidation, it is yellow
by transmitted light. Such a bead is sensitive in a very high
degree, quite as much so as canary glass. When the light falls
sideways on it, and it is held against black cloth or a dark object,
it exhibits plainly the green colour due to internal dispersion.
When properly examined by means of sunlight its sensibility is
evident at once, and when the dispersed light is viewed through a
prism it is resolved into bright bands. One of the most convenient
modes of examining such minute objects consists in reflecting the
sun’s light horizontally through a large lens, intercepting by
means of absorbing media all the rays except those of very high
refrangibility, placing the object to be examined in the condensed
beam, and viewing it through a prism. So delicate is this test
when applied to uranium, that on one occasion, when engaged in
examining a bead coloured green by chromium, which had been
fused in the exterior flame, I observed the appearance given by
uranium. This turned out to be actually due to uranium, of

See note D.

350

ON THE CHANGE OF REFRANGIBILITY OF LIGHT.

which a mere trace was accidentally present without my know¬
ledge.

160. The green communicated to microcosmic salt by uranium
after exposure to the reducing flame has a very peculiar com¬
position, by means of which the presence of uranium may be
instantly detected. For this purpose it is sufficient to view
through a prism the inverted image of the flame of a candle
formed by the bead, the latter being so held as to be seen
projected on a dark object. The observation is perfectly simple,
and occupies only a few seconds. The spectrum exhibits an
isolated band at the red extremity, followed by a very intense
dark band of absorption. A similar dark band, but not quite so
intense, occurs in the green: beyond the green there is usually
but little light seen. As the absorption progresses the first dark
band invades all the space from the red to the green, and the
spectrum consists of an isolated red band and a green band
divided into two. In its mode of absorption, the medium has a
strong general revsemblance to chlorophyll. The green due to
copper or to chromium shows nothing remarkable when viewed
through a prism, and could not possibly be confounded with the
green due to protoxide of uranium. The absorption bands due
to this oxide are not completely brought out till the bead is
cold.

161. Uranium produces the same effects with borax as with
microcosmic salt, but they are less distinct, or at least less easily
produced.

162. When the uranium contained in a bead of microcosmic
salt is thoroughly oxidized, and the bead is gently heated, so as
just to be self-luminous, the light which it gives out is not red,
like that of most substances at a low heat, but green, or rather
greenish white.

163. Solutions of protoxide of uranium have a very remark¬
able effect on the spectrum, resembling more or less that of a
bead of microcosmic salt coloured green by uranium. Of course
the absorption can be observed much better by means of a solu¬
tion than by a mere bead. I have observed several bands of
absorption in such solutions, but the cases which I have hitherto

ON THE CHANGE OF EEFRANGIBILITY OF LIGHT.

351

examined are too few to justify me in entering into detail.
Besides, the absorption bands due to protoxide of uranium do not
belong properly to my subject, the compounds of this oxide, so
far as I have examined, being insensible.

Appearance of highly Sensitive Media in a Beam from which the
Visible Rays are nearly excluded.

164. When a large beam of sunlight is reflected horizontally
into a darkened room, and transmitted through an absorbing
medium, placed in the window, of such a nature as to let pass
only the feebly illuminating rays of high refrangibility and the
invisible rays beyond, various sensitive media have a very strange
and unnatural appearance when placed in the beam, on account
of the peculiar softness of the dispersed light with which the
media appear as it were self-luminous, and the almost entire
absence of strong light reflected from convexities. Among sub¬
stances eminently proper for this experiment, may be mentioned
a solution of the bark of the horse-chestnut*', or of sulphate of
quinine, or of stramonium seeds, a decoction of madder in a solu¬
tion of alum, and above all, ornamental articles of canary glass.
The appearance of a specimen of yellow uranite was curiously
altered by this mode of examination. By daylight the mineral
appeared much of the same colour as the stone in which it was
imbedded, but when placed in a beam such as that above men¬
tioned the uranite was strongly luminous, while the stone re¬
mained dark.

Natural Crystals.

165. Of natural crystals I have hitherto examined only a
small number. For a long time I was occupied almost exclusively
with vegetable products, the mineral kingdom not appearing
promising. However, I have found internal dispersion in certain
specimens of apatite, aragonite, chrysoberyl, cyanite, and topaz.
In ail these cases the dispersion appeared due, as in the case of

[* A solution which answers admirably, and is very easily prepared, is obtained
by adding to a decoction when cold a suitable quantity of alum or a ferric salt,
precipitating by ammonia, and filtering. The powerful fluorescence of the
solution is due to mixed aesculin and fraxin.]

352

ON THE CHA.NGE OF REFR ANGIBILITY OF LIGHT.

fluor-spar, to some substance accidentally present in small quan¬
tity; so that yellow uranite is at present the only natural crystal
to the essential constituents of which the property of internal
dispersion has been found to belong.

166. Among the minerals just mentioned apatite was the
most sensitive, though it fell very far short of yellow uranite.
That the sensibility was not due to phosphate of lime, was plain
from the circumstances that a colourless specimen was insensible,
and that the amount of sensibility was found to be different in
different parts of the same sensitive specimen. With the excep¬
tion of the colourless crystal already mentioned, all the specimens
of apatite examined were of a greenish colour, and all were sensi¬
tive. The dispersed light was something of an orange colour,
but was not homogeneous orange. In one specimen it consisted
of three distinct bright bands at regular intervals. The mode in
which the sensibility of this crystal was connected with the
refrangibility of the incident rays was very peculiar. In ara¬
gonite dispersion was found in the transparent specimens examined;
the translucent specimens were found to be insensible. The dis¬
persed light was of a brownish white colour. In the same crystal
some parts were insensible and others more or less sensitive. The
portions of equal sensibility were arranged in plane strata, just
as in the case of fluor-spar, as has been noticed by Sir David
Brewster. In a specimen which had been cut for showing conical
refraction, the strata were in some places perpendicular to the
plane of the optic axes, and in other parts parallel to the line
bisecting the axes, and inclined to their plane at such an angle
that the two directions of the strata must have been parallel to
two of the commonest lateral faces. Another specimen showed
strata parallel to an oblique terminal face. The strata are plainly
due, as Sir David Brewster has remarked with reference to fluor¬
spar, to some substance taken up during crystallization. Accord¬
ingly, they preserve a sort of history of the growth of the crystal.
In a twin crystal of fluor-spar, the direction of the strata in that
part of the mass which was common to the geometrical forms of
both crystals, showed to which crystal it really belonged. In
fluor-spar the strata are parallel to the faces of the cube, at least
in the specimens which I have examined, and the same has been
observed by Sir David Brewster.

ON THE CHANGE OF REFRANGIBILITY OF LIGHT. 353

In chrysoberyl, cyanite and topaz, the dispersed light was red
or reddish, and was too variable to allow of its being attributed to
the essential constituents of the crystals. In these cases the
sensibility was but slight; indeed in cyanite there was only a
trace of dispersion when the crystal was examined under great
concentration of light.

Coloured Glasses.

167. Besides canary glass, I have examined the common
coloured glasses, including that coloured by gold, but with one
exception have not met with any example in which the sensibility
observed appeared to have any connexion with the colouring
matter. The paler glasses exhibited a little internal dispersion,
because the colour was not sufficiently intense to mask the disper¬
sion which a common colourless glass would exhibit.

168. The exception occurred in the case of the pale brown
glass, which has been already mentioned in connexion with my
first experiment. This glass dispersed a red light under the
influence of the highly refrangible rays. The colour of the light
was not pure prismatic red, but red was predominant. A similar
dispersion, due apparently to the same cause, was observed in the
case of one of the common reddish brown German wine bottles.
The sensibility of these glasses appears to be due to an alkaline
sulphurct. [?] A bead purposely coloured in this manner was in fact
found to disperse a red light like the glasses. Moreover, in the
confused masses obtained by fusing sulphate of soda and sulphate
of potash on charcoal before the blowpipe, certain portions were
found which dispersed a red light, and that pretty copiously for
an inorganic substance. A similar dispersion was observed among
the products obtained by fusing together sulphur and carbonate
of potash, while other parts of the confused mass exhibited disper¬
sion of a different kind. It seems plain that among the combina¬
tions of sulphur with the alkalies sensitive compounds exist, but
what they are I have not examined.

s. ill.

23

354

ON THE CHANGE OF REFRANGIBILITY OF LIGHT.

Cautions with respect to the discrimination between true and
false internal dispersion.

169. In the early part of this paper certain tests were given
for distinguishing between true and false internal dispersion in a
fluid. But it requires some experience in observations of this
kind to be able readily to decide, and a too rigid adherence to
one of the tests to the exclusion of the others might lead to error.

The first test relates to the continuous appearance of a truly
dispersed beam. But sometimes solid particles exist in mechanical
suspension, which are so fine and so numerous, that this test alone
might lead the observer to mistake a falsely for a truly dispersed
beam. On the other hand, if a fluid which itself alone exhibits
no internal dispersion, true or false, hold solid particles in what is
obviously mere mechanical suspension, we must not immediately
conclude that the medium, taken as a whole, is incapable of
changing the refrangibility of any portion of the light incident
upon it. For we have seen that the fluid state is not in the least
degree essential to the exhibition of sensibility, and of course a
fluid will serve as well as anything else for the mere mechanical
support of a sensitive substance.

170. Thus lycopodium is very sensitive, as appears by examin¬
ing the powder in a linear spectrum. Accordingly, I found that
when a little lycopodium was mixed with water, and the whole
medium was examined by the fourth method, it displayed its
sensibility, although the beam of light which had changed its
refrangibility was plainly discontinuous. When Indian yellow
was used instead of lycopodium, the whole medium exhibited its
sensibility when it was examined by the fourth method. In this
case the suspended particles were so fine that the beam of light
which had changed its refrangibility appeared to be continuous,
though of course it was not really so. In observing with muddy
fluids like these, it is almost necessary to employ absorbing media,
since otherwise the effect of the light scattered at the surfaces of
the prisms and large lens might lead the observer to conclusions
altogether erroneous.

171. The next test relates to the polarization of a falsely
dispersed beam. Being engaged on one occasion in examining

ON THE CHANGE OF REFRANGIBILITY OF LIGHT.

355

the effects of acids and alkalies on a weak solution of a sensitive
substance, employing sunlight which had been merely reflected
through a small lens, I met with a beam which had every appear¬
ance of having been only falsely dispersed, but on viewing it from
above through a doubly refracting prism I was surprised at first
by finding it unpolarized. It soon occurred to me that the beam
must have been due, not to solid motes, but to excessively small
bubbles of carbonic acid gas, the existence of which was thus
revealed, though they were too small to be seen directly. The
light being incident on these bubbles at an angle of about 45°,
which is very little less than the angle of total reflexion, the
reflected light would be almost perfectly unpolarized*.

172. Water which had been merely boiled in a test tube gave
a similar result. The unpolarized beam of falsely dispersed light
was of course due in this case to the air which had been held in
solution. This shows why long-continued boiling should be neces¬
sary, in order to free water from air. It is not that the affinity of
water for air is so great as to be only gradually overcome, but
that the air, immediately expelled from solution when the tempe¬
rature rises sufficiently, is still retained in a state of mechanical
mixture, forming excessively minute bubbles, the terminal velocity
of which is insensible. Accordingly it is not till larger bubbles
are formed, by the casual meeting of a number of these small
bubbles, that the air rises to the surface and escapes.

173. With respect to the test of true dispersion depending on
the change of refrangibility, it has been already remarked that in
some cases the change is so slight, that if this test alone were
applied, the observer might mistake true dispersion for false.
However, it is only in rare cases that there is any danger of being
deceived in this manner in the application of the test; but on the
other hand, in observing a muddy fluid or a translucent solid by
the fourth method, the observer, if not on his guard, might easily
be deceived by the effect of scattered light, and be led to mistake
false dispersion for true. Thus suppose the medium to be water
holding in suspension particles of an insensible water colour, and
the small lens to be placed a little beyond the commencement of
the violet. Two beams of light would enter the lens, namely, a

* See note E.

23—2

356 ON THE CHANGE OF REFRANGIBILtTY OF LIGHT.

regularly refracted beam of violet, and a scattered beam of white
light. Of these the latter would be insignificant compared with
the former, were it not that the illuminating power of the colours
belonging to the middle of the spectrum is so very much greater
than that of the violet. When the dispersed beam was analysed
by a prism, it would be decomposed into a violet beam of definite
refrangibility, followed by a dark interval, and then a broad band
containing the colours of the brighter part of the spectrum in
their natural order. This is what is constantly seen in cases of
true dispersion; but the polarization of the beam, and its beha¬
viour under the action of absorbing media, would reveal the coun¬
terfeit character of the dispersion.

On the Colours of Natural Bodies.

174. By this expression I mean to include only fin* colours to
which it is usually applied, namely, those of leaves, {lowers, paints,
dyed articles, &c., which form the great mass of the colours that
fall under our observation. I do not refer to colours duo to refrac¬
tion, such as those of the rainbow, or to diffract ion, such as those
of the corona*, seen about the sun and moon, or to interference, such
as those seen in the clear wings of small flies, or to the colours
which accompany specular reflexion, which last arc usually but
slight, though sometimes pretty intense.

In some few instances, as for example in tin* case of fluor-spar,
various salts of peroxide of uranium, acid solutions of disulphale of
quinine, &c., colours are observed, sufficiently strong to arrest, at¬
tention, which have a remarkable and hitherto unsuspeete<I origin.
But I am not now speaking of colours arising from a, change of
refrangibility in the incident light. In the vast, majority of eases
these colours are far too feeble, to form any sensible, portion of
the whole colour observed. The colours which dyed articles give,
out under the influence of the highly refrangible rays usually
agree more or less nearly with those of which such substances
commonly appear, and it is possible that the colour arising from a,
change of refrangibility may contribute in some slight degree to
the brilliancy of the tint observed. If, however, tin* effect ho
sensible I am persuaded that it is but slight; and very brilliant
colours may be produced without a change of refrangibility, as for

ON THE CHANGE OF RE FRA N GIBILIT Y OF LIGHT. 357

example in the case of biniodide of mercury. For the present I
shall neglect the light which may have changed its refrangibility.

175. Few, I suppose, now attach much importance to the bold
speculations in which Newton attributed the colours of natural
bodies to the reflexion of light from thin plates. Sir David
Brewster has shown how extremely different the prismatic com¬
position of the green of the vegetable world is, from what it ought
to be, according to Newton's theory, and what Newton supposed
that it was. It is now admitted that the various colours of natural
bodies are merely particular instances of one general phenomenon,
namely, that of absorption. Absorption is most conveniently
studied in a clear fluid or solid, but it does not the less exist in a
body of irregular structure, such as a dyed cloth or a coloured
powder.

The green colouring matter of leaves affords an excellent
example of the identity of the effect produced on light by natural
bodies and of ordinary absorption; for the same very peculiar
system of absorption bands which are displayed by a clear solution
of the colouring matter may be observed directly in the leaf itself.
However, it is needless to bring forward arguments to support a
theory now I suppose universally admitted; my present object is
merely to point out the mode in which the colours which bodies
reflect, or more properly scatter externally, depends upon the
absorbing power of the colouring matter, so as to justify the conclu¬
sions deduced in Art. 142, from observations made in the manner
there described.

176. Let white light be incident on a body having an irregular
internal structure, such as a coloured powder. A portion will be
reflected at the first irregular surface, but the larger portion will
partly enter the particles, partly pass between them, and so
proceed. In its progress the light is continually reflected in an
irregular manner at the surfaces of the particles, and a portion of
it is continually absorbed in its passage through them, 1 or
simplicity's sake, suppose the light incident in a direction perpen¬
dicular to the general surface, and neglect all light which is more
than once reflected. Let t be the thickness of a stratum which the
light has penetrated, I the intensity of the light at that depth, or
rather the intensity of a given kind of light, so that the whole

358

ON THE CHANGE OF RKFRANGIBIUTY OF LIGHT.

intensity may be represented by fld/i, p being the refractive index
in some standard substance. In passing across the stratum whose
thickness is dt 9 suppose the fraction qdt of the light to be absorbed,
and the fraction rdt to be reflected and scattered in all directions,
then

dl = — {q + r) Idt

Integrating this equation, and supposing 7 U to be the initial value
of /, when t = 0, we have

1 = 1#-®+* ...(a).

For the sake of simplicity, suppose the body viewed in a
direction nearly perpendicular to the general surface; and of the
light reflected and scattered in passing across the stratum whose
thickness is dt, suppose that the fraction n would enter the eye if
none were lost by absorption, &c. Then thi* intensity of the light
coming from that stratum would be nrhlt . .But, in getting hack
across the stratum whose thickness is /, the intensity is diminished
in the ratio of I 0 to I. Hence if I' be the intensity of the light
actually entering the eye,

dl 1 = nrJ^lh/t = nrl u e^' ,lkrit dt

If we suppose the thickness of the body suflieient to develope
all the colour which the body is capable of giving, the superior
limit of t will be co , and we shall have

177. The colour which accompanies ordinary reflexion being
usually but slight, I shall neglect tint chroma,tic variations of r.
It is q which is subject to extensive and apparently eaprieious
variations, depending upon the. refrangibility of the light. Imagine
two curves drawn whose abscissa* are proportional to /g and
ordinates proportional to the ratio of / to /„ for the firsthand the
ratio of I' to I Q for the second. These curves will serve to repre¬
sent to the mind the composition of the light transmitted through
a stratum of the body having a thickness t, and of that reflected
from the body when seen in mass. It is plain that t he maximum
and minimum ordinates in the two curves will correspond to the
same abscissae; but unless t be very small, so small as to be
insufficient to bring out the colour of the medium seen by trails-

ON THE CHANGE OF REFRANGIBILITY OF LIGHT. 359

mission, the maxima and minima will be much more developed in
the first curve, whose ordinates vary as e~^ } than in the second,
whose ordinates vary as (q + r)" 1 . If, then, the absorbing power
be subject to fluctuations depending on the refrangibility of the
light, the bands of absorption may be observed either in the
reflected or in the transmitted light, but they admit of being
better brought out in the latter.

178. If the nature of the substance be given, q will be given.
If now the body be of a loose nature, as for example blue glass
reduced to a fine powder, r will be considerable. Hence, in
accordance with the expression (b), the quantity of light scattered
externally will be considerable, but the tint will be but slight. If
the powder be now wetted, the reflexions at the surfaces of the
particles will be diminished, r will be diminished, and, as appears
from (b), the quantity of light scattered externally will be di¬
minished, but at the same time the tint will be deepened, since
the chromatic variations of F are increased. If the body be
compact and nearly homogeneous, r will be small, and therefore
very little light will be returned, except what is regularly reflected
at the first surface. The tint of the small quantity of light which
is reflected otherwise than regularly, will be somewhat purer than
before, inasmuch as the chromatic variations of I' tend to become
the same as those of q~ l .

On the nature of False Dispersion , and on some
applications of it.

179. Tt has been already stated that a beam of falsely
dispersed light seen in a fluid has generally more or less of a
sparkling appearance, indicating that it owes its origin merely to
motes held in mechanical suspension. Sometimes, however, no
defect of continuity is apparent. This is especially the case when
two fluids arc mixed together, of which one contains in solution a
very small quantity of a substance which we might expect to be
precipitated by the addition of the other, or when a slightly
viscous fluid has remained quiet for a long time. If some part at
least of a falsely dispersed beam be plainly due to motes, that
does not of course prove for certain that there is no part which
may have a different origin, and may be essentially connected with

360

ON THE CHANGE OF RKFUANUl BILITY OF LIGHT.

true dispersion; nor do the theoretical views which I entertain of
the cause of the latter lead me to regard it as at all impossible
that a beam polarized in the 1 plant 1 of reflexion, and having the
same refrangibility as the incident light, may bo a necessary
accompaniment of true dispersion. However, observation, I think,
points in a contrary direction; for although more or less of false
dispersion is almost always exhibited along with true dispersion,
the quantity of the former seems to have no relation to the
quantity of the latter, but does seem to have relation to the
greater or less degree of clearness which we should hi 1 disposed to
attribute to the fluid.

180. The phenomenon of false internal dispersion seems to
admit of being applied as a chemical test to determine whether or
not precipitation takes place. Thus, if a little tincture of turmeric
be greatly diluted with alcohol, and then water be added, a yellow
fluid is obtained which appears to he perfectly clear, exhibiting no
sensible opalescence; but the occurrence of’a copious false dispersion
when the fluid is examined by sunlight, reveals at once, the
existence of suspended particles, though they are too minute to he
seen individually, or even to give a diseonfinuous appearance to
the falsely dispersed beam. Although such a precipitation could
not, I suppose, be used as a means of mechanical separation, it
might still be useful as pointing out tin* possibility of an actual
separation under different circumstances as to strength of solu¬
tion, &c.

181. One of the best instances of falsi* dispersion that I have*
met with, best, that is, in forming a most excellent imitation of
true dispersion, occurred in tin* ease of a specimen of plate-glass
which was made, as 1 was informed, with a quant it v of alkali barely
sufficient. This glass, which was very slightly yellowish brown,
when viewed edgeways by transmitted light, had a, bluish appear¬
ance when viewed properly, strongly resembling that of a decoction
of the bark of the horse-chestnut, diluted with water till the
dispersed light is no longer concentrated in tin* neighbourhood of
the surface. But when the glass was examined by sunlight, the
polarization of the dispersed beam, and the identity of its refrangi-
bility with that of the incident light, showed that this w; is merely
an instance of false dispersion. Another very good example of

ON THE CHANGE OF REFRANGIBILITY OF LIGHT.

361

false dispersion is afforded by chloride of tin dissolved in a very
large quantity of common water.

182. When a horizontal beam of falsely dispersed light is
viewed from above, in a vertical direction, and analysed, it is found
to consist chiefly of light polarized in the plane of reflexion. It
has often struck me, while engaged in these observations, that
when the beam had a continuous appearance, the polarization was
more nearly perfect than when it was sparkling, so as to force on
the mind the conviction that it arose merely from motes. Indeed,
in the former case, the polarization has often appeared perfect, or
all but perfect. It is possible that this may in some measure have
been due to the circumstance, that when a given quantity of light
is diminished in a given ratio, the illumination is perceived with
more difficulty when the light is uniformly diffused than when it
is spread over the same space, but collected into specks. Be this
as it may, there was at least no tendency observed towards polariza¬
tion in a plane perpendicular to the plane of reflexion, when the
suspended particles became finer, and therefore the beam more
nearly continuous.

183. Now this result appears to me to have no remote bearing
on the question of the direction of the vibrations in polarized light.
So long as the suspended particles are large compared with the
waves of light, reflexion takes place as it would from a portion of
the surface of a large solid immersed in the fluid, and no conclusion
can be drawn either way. But if the diameters of the particles be
small compared with the length of a wave of light, it seems plain
that the vibrations in a reflected ray cannot be perpendicular to the
vibrations in the incident ray*. Let us suppose for the present,

[* xiio way in which X at the time regarded the problem was as follows.
Suppose polarized light to he passing through a medium which holds in suspension
a vast number of excessively fine particles of some substance different from the
medium itself, the dimensions of the particles being for simplicity supposed
extremely small compared with the length of a wave. The ether in the medium
will ho vibrating to and fro in a direction perpendicular to the direction of
propagation, and cither in or perpendicular to the plane of polarization. The
inertia of the particles being presumably very great compared with that of a
corresponding volume of the ether alone, the ponderable particles may be supposed
to remain at rest, and they will therefore disturb the motion of the ether, and
cause vibrations to spread out from them in the ether. Now the repose of the
particles may be regarded as the resultant of two equal and opposite motions, one

362 ON THE CHANGE OF RKFRANGIBIUTY OF LIGHT.

that in the case of the beams actually observed, the suspended
particles were small compared with the length of a wave of light.
Observation showed that the reflected ray was polarized. Now all
the appearances presented by a plane-polarized ray are symmetrical
with respect to the plane of polarization. Hence wo have two
directions to choose between for tin* direction of the vibrations in
the reflected ray, namely, that of the incident ray, and a direction
perpendicular to both the incident and the reflected rays. The
former would be necessarily perpendicular to the directions of
vibration in the incident ray, and therefore we are obliged to
choose the latter, and consequently to suppose that tin* vibrations
of plane-polarized light are perpendicular to tin* plane of polariza¬
tion, since experiment shows that the plane of polarization of the

the same as that of the ether itself, the other a to and fro motion along the. same
line as the former but in tin* opposite direction ; and we may superpose these
motions as regards their effect on the ether. In the former the pariides would he
moving with the ether, and therefore* would not disturb it; as r< gaids the latter we
may think of the particles as moving to and fro in otloruhe * till ether, and
producing therefore an ethereal disturbance emanating in all directions from the
particle. This disturbance having to be transversal will evidently be ml in a
polar direction and a maximum in an cquatoiml direction, \aiying in fart in
amplitude as the sine of the polar distance, the polar line bring a line through flu*
middle of the particle drawn in the direction of Hu* incident vibrations. 'I’lu*
direction of propagation of the incident light and that of the line of l ight being an
in the text, if the incident light be common light, we may irplaee it by two
independent streams, of equal intensity, pohuized the one in a vertical and the
other in a horizontal plane; and of these the one f.»r which the plane of vibration
is vertical will not give rise to any diffracted light entering the eye, while the
other will give rise to a stream for winch the direction of \ibrafion i» horizontal,
and which is therefore polarized in such a manner that the plane of \ihration
passes through the line of sight and is perpendicular to the din-choii of propaga¬
tion of the incident light, and which therefoic may he ext ingui.4u*d bv an annhser
suitably turned; and in a similar way, as stated in the text, the light entering the
eye may be quenched by polarizing the light before incidence on the pai tides
instead of analysing it after diffraction. The <•(niche ions of theory, uhirh are
enunciated with reference to tin* plane of vibration, exactly agree with the results
of experiment, which are described with reference to the plane of polarization;
and to make the two lit we must suppose, the direction of vihiation in polarized
light to be perpendicular, not parallel, to tin* plane of polm i/,ation.

.there can be little doubt that in several cases which fell under my notice,
especially in that of the glass mentioned in Art. IK1, the particles wore sulliciently
fine to lender the above reasoning applicable. Still more must that have been
the case in the beautiful experiments ot Tyndall on the decomposition of gases
and vapours by rays of high retrangihility, who was led independently to the same
conclusions as those stated in tin* text regarding the phenomena of polarization
exhibited by fine particles in suspension.J

ON THE CHANGE OF REFRANGIBILITY OF LIGHT. 363

reflected ray is the plane of reflexion. According to this theory, if
we resolve the vibrations in the incident ray horizontally and
vertically, the resolved parts will correspond to the two rays,
polarized respectively in and. perpendicularly to the plane of
reflexion, into which the incident ray may be conceived to be
divided, and of these the former alone is capable of furnishing a
reflected ray, that is of course a ray reflected vertically upwards.
And in fact observation shows that, in order to quench the
dispersed beam, it is sufficient, instead of analysing the reflected
light, to polarize the incident light in a plane perpendicular to the
plane of reflexion.

Now in the case of several of the beams actually observed, it is
probable that many of the particles were really small compared
with the length of a wave of light. At any rate they can hardly
foil to have been small enough to produce a tendency in the
polarization towards what it would become in the limit. But no
tendency whatsoever was observed towards polarization in a plane
perpendicular to the plane of reflexion. On the contrary, there did
appear to be a tendency towards a more complete polarization in
the plane of reflexion.

M. Babinct has been led by the same reasoning to an opposite
conclusion respecting the direction of the vibrations in polarized
light, resting on an experiment of M. Arago’s, in which it appeared
that 'when light was incident perpendicularly on the surface of
white paper, and the reflected or rather scattered light was viewed
in a direction almost grazing the surface, it was found to be
partially polarized in the plane of the sheet of paper*. But the
actions which take place when light is incident on a broad irregular
surface, like that of paper, bounding too a body which is so trans¬
lucent that a great part of the light must enter it and come out
again, appear to me to be too complex to allow us to deduce any
conclusion from the result respecting the direction of vibration.
Besides, the result itself admits of easy explanation, by attributing
it to the light which has entered the substance of the paper and
come out again, which might be expected to be polarized by

refraction.

* Comptes liendus, Tom. xxix. p. 514.

364 ON THE CHANGE OF REFRANGIBILITY OF LIGHT.

Effect of Heat on the Sensibility of Glass, <tc.

184. The sensibility of glass is temporarily destroyed by heat.
The glass may be heated by holding it in the flame of a spirit-lamp,
as a heat much short of redness is sufficient. This takes place
even with glass coloured by oxide of uranium, which is in general
so highly sensitive. The sensibility returns again as the glass
cools. A bead of rnicrocosmic salt, containing uranium in its
highest state of oxidation, is very sensitive when cold, but
insensible when hot. The sensibility gradually comes on as the
bead cools. A solution of nitrate of uranium in water on being
heated has its sensibility impaired, very much so by the time the
temperature reaches the boiling-point. The sensitive compounds,
whatever may have been their precise nature, obtained by fusing
the sulphates of soda and potassa on charcoal before the blowpipe,
were insensible while hot. The few vegetable solutions which I
have examined with this object did not seem to have their
sensibility affected by being heated.

Effect of Concentration and Dilution.

185. In investigating the change of refrangibility produced
by a sensitive substance in solution, it is almost always convenient
to have the solution weak. This however is by no means merely
a matter of convenience, for the quantity of light which the
medium is capable of giving back with a changed refrangibility
is often materially diminished by increasing the concentration
of the solution. Thus a solution which, when in a concentrated
state, exhibits no sensible dispersive reflexion, will often exhibit
when much diluted a very copious appearance of that nature.
On the other hand, the dilution may of course lx* carried too
far, so as to render imperceptible the peculiar properties of the
substance dissolved. Yet it is wonderful what a degree of di¬
lution a highly sensitive solution will bear before its sensibility
ceases to be perceptible.

That the sensibility will be diminished, and will at last
become imperceptible, if only the dilution he carried far enough,
is nothing more than might have been predicted with the utmost
confidence. In such a case the light passes completely through

ON THE CHANGE OF REFRANGIBXLITT OF LIGHT. S65

the fluid long before it has produced all the effect which it is
capable of producing. But that concentration should be an
obstacle to the exhibition of the phenomenon is not perhaps what
we should have expected, and deserves an attentive consideration.

186. Imagine a given sensitive substance to be held in
solution, in a vessel of which the face towards the eye is plane,
and the breadth in the direction of vision as great as we please;
and suppose the solvent, or at least the fluid used for diluting the
solution, to be itself colourless and insensible. Suppose the fluid
to be illuminated by light of given intensity and given refran-
gibility entering at the face next the eye, and let the eye E
from a given position look in the direction of a given point P
in the nearer surface of the vessel. In short, let everything be
given except the strength of the solution. For the sake of
simplicity regard the eye as a point, and make E the vertex
of an indefinitely thin conical surface surrounding the line EP.
Call this conical surface C, and let c be the surface within the
fluid generated by right lines coinciding with the refracted rays
which would be produced by incident rays coinciding with the
generating lines of the surface C. This latter surface we may
if we please regard as cylindrical, since we shall only be concerned
with so much of the fluid contained within it as lies at a distance
from P less than that at which the light entering the eye in
consequence of internal dispersion ceases to be sensible; and in
the cases to which the present investigation is meant to apply
this distance is but small compared with PE. Let the fluid
within c be divided into elementary portions by planes parallel
to the surface of the fluid at P, and at distances from P pro¬
portional to the strength of the solution. It is evident that an
element of a given rank, reckoned from P, will contain a constant
number of sensitive molecules, and the incident light in reaching
this element has to pass through a thickness of the medium such
that a plate of the same thickness, and having a given area,
contains a given number of sensitive or absorbing molecules.
The same is true of the dispersed light which proceeds from
the element and enters the eye. Now it seems natural to suppose
that if the strength of a solution be doubled, trebled, &c., or
reduced to one-half, one-third, &c., the quantity of light absorbed
will bo the same provided the length of the path of the light be

366

ON THE CHANGE OF REFBANfHBILITY OF LIGHT.

reduced to one-half, one-third, &c., or doubled, trebled, &c. This
comes to the same thing as supposing that each absorbing mole¬
cule stops the same fractional part of the light passing it, whether
the solution be more or less dilute. We should similarly be
inclined to suppose that each sensitive molecule would give out
the same quantity of light, when influenced by light of given
intensity, whether it belonged to a stronger or a weaker solution.
If we admit these suppositions, it is plain that the quantity of
dispersed light which reaches the eye. from the element under
consideration will be independent of the strength of the solution.
This being true for each element in particular will be true for
the aggregate effect of them all, and therefore the quantity of
light exhibited by dispersive reflexion will be independent of the
strength of the solution. It may he readily seem that the result
will be the same if we take into account the finite* size* of the
pupil

187. Now this is by no means true* in (*xpi*rim<mt. On
examining in a pure spectrum a highly concent rated solution of
sulphate of quinine, a copious dispersion was observed to
commence a little below the fixed line* (}. It remaine*d ve*ry
strong as far as II, and beyond. In the* weak solutiem first
mentioned in this paper, it will l)c* remembered that Urn dis¬
persion seemed to come on about Q\IL The mason of this, or
at least one reason, is evident, and was vt*ry prettily shown by
the form of the space to which the*. disperse*d light was confined.
On looking down from above*, so that this spare* was srem in
projection, it appeared in the* case of the* we*ak solution to have
approximatedy the form of the spare* contained bet with one*,
branch of a rectangular hyperbola, one* asymptote, and a line
parallel to the other, thei first, as\mpt.otii l><*ing the* projection
of the anterior surface, and thei line* paraded to the* other being
the course of the least refrangiblii of thei aetivei rays whiedi we*re;
capable of producing a sensibhi epiantify of dispensed light,. Tim
breadth of the illuminated spae*.e, wliieh among the* most, highly
refrangible rays was almost insemsible, continually incivased, until
the space ended in a blue beam which wemt quite across the
vessel. But in the case of the strong solution thei illuminated
space had throughout an almost insensible breadth, except, just
close to its lower limit, that is, the limit corresponding to the

ON THE CHANGE OF REFRANGIBILITY OF LIGHT. 367

least refrangible of the active rays, where it ended in a sort of
tail or plano-concave wedge, which penetrated to a moderate
distance into the fluid. Hence one reason, though perhaps not
the only reason, why the strong solution showed a copious dis¬
persion from G to G^H, where the weak solution showed hardly
any, is plain enough. But in the region of the invisible rays
beyond the violet, the dispersion was plainly more copious with
the weak than with the strong solution. It appears then that
in such a case the sensitive molecules do not act independently
of each other, but the quantity of light emitted by a given
number of molecules is less, in proportion to -the light (visible or
invisible) consumed, than when a solution is more dilute. We
should expect a priori that when a solution is tolerably dilute
further dilution would make no more difference in this respect.
This seems to agree very well with experiment. For when a
pretty dilute solution and one much more dilute are compared
with respect to the quantity of dispersed light given out in a
given portion of the incident spectrum, they appear to be alike.

I suppose the comparison to be made with respect to such a
portion of the incident spectrum, or in the case of solutions of
such strength, that the dispersed light is confined to a space
extending to no great distance into the fluid in either solution.
Under these circumstances the comparison may be made easily
enough.

188. Tn the actual experiment, the elementary portions of
light coming from the elementary strata of fluid situated at
different distances from the anterior surface enter the eye to¬
gether. Let us however trace the consequences of the very
natural supposition, that in passing across a given stratum of
fluid the quantity of light absorbed, as well as the quantity
given out by dispersion, is proportional, cceteris paribus , to the
intensity of the incident light. The incident light is here sup¬
posed to be homogeneous, and to belong indifferently to the
visible or invisible part of the spectrum. In crossing the
elementary stratum having a thickness dt, let the fraction qdt of
the incident light be absorbed, and the fraction rdt dispersed in
such a direction as to reach the eye; and of the latter portion
let the fraction sdt be absorbed in crossing a stratum having a
thickness dt, s being different from q on account of the change

368 ON THE CHANGE OF REFRANGIBILITY OF LIGHT.

of refrangibility. Then by a very simple calculation similar to
that of Art. 176, we find for the intensity /' of the dispersed light
which enters the eye

r = -— / 0 ,

q + s

I 0 being the intensity of the incident light. Since a sensitive
fluid is in general coloured, and the dispersed light is in general
heterogeneous, s will in general be different for the different
portions into which the dispersed light would be decomposed by
a prism. However, if the fluid be colourless, or all but colourless,
as is the case with a solution of sulphate of quinine, ,v will be
insensible, so that T will be proportional simply to r<f\ Hence
from the observed variations in arising from variations in the
strength of the solution, we may infer the corresponding varia¬
tions in ref 1 .

If, then, we represent by the ordinate of a curve the ratio of
the quantity of light given out to the quantity of light absorbed
by a given number of active molecules, the abscissa being the ratio
of the quautity of diluting fluid to the quantity of tin 1 sensitive
substance in solution, it appears that the curve will be concave
towards the axis of the abscissa 4 , and will have an asymptote
parallel to that axis.

On the Choice of a Screen.

189. We have seen that white paper, the substance com¬
monly employed as a screen on which to receive tin; spectrum,
gives back with a changed refrangibility a portion of tint light
incident upon it. This might in some cases lead an observer not
aware of the circumstance to erroneous conclusions. Since the
colour of dispersed light depends upon its refrangibility, which is
different from that of the active light, the colours of a, spectrum
received on white paper must be somewhat modified. In truth
the intensity of the light dispersed is so small compared with the
intensity of the light scattered, that the modification is quite
insensible except in the extreme violet. But beyond the extreme
violet the spectrum seems to be prolonged with a sort of greenish
gray tint, which belongs neither to that nor to any other part of
the true spectrum. In experiments on absorption, if instead of

ON THE CHANGE OF REFRANGIBILITY OF LIGHT. 369

receiving the light directly into the eye it be found convenient to
form a pure spectrum ou a screen of white paper, then, if the
absorbing medium be placed in the path of the incident light, the
scattered light forming any part of the spectrum cannot be cut off
or weakened without at the same time cutting off or weakening
the dispersed light coming from the same part of the screen.
But if the absorbing medium be held in front of the eye, its effect
on the spectrum will sometimes be very sensibly different from
what it would be were the screen to send back none but scattered
light.

It is true that the quantity of light dispersed by white paper
is so small that this substance may very well continue to be used
as a screen, without any danger of the observer’s being deceived, if
only he be aware of the fact of dispersion, so as to be on his guard.
Still, it is not unreasonable to seek for a substitute for paper,
which may be free from the same objection.

190. A porcelain tablet appeared to be unexceptionable in
this respect, for it exhibited no perceptible sensibility, even when
examined by a linear spectrum. However, the translucency of the
substance gave the spectrum a blurred appearance, and the fixed
lines were not shown so well as on paper.

Chalk scraped smooth is well adapted, from its fineness, its
whiteness and its opacity, for showing the most delicate objects.
The finest fixed lines are beautifully seen on it, decidedly better
than on paper. Its sensibility too, though not absolutely null, is
much less than that of most kinds of white paper. Indeed, it
would be an unnecessary refinement to seek for anything better,
were it not that a piece of sufficient size might not always be at
hand. From what I have seen, I believe that the best kind of
screen will be obtained by the use of some white inorganic
chemical precipitate, but my experiments in this department have
not yet been sufficiently extended to authorize me in recommend¬
ing any particular process.

191. The object of the observer may however be altogether
different, and he may wish to extend the spectrum as far as possi-
ble, for the purpose of viewing the fixed lines belonging to tbe
invisible part beyond the extreme violet, or making experiments
on the invisible rays. For this purpose it would be proper to

24

s. III.

370 ON THE CHANGE OF KEFRANGIBILITY OF LIGHT.

employ a clear and highly sensitive solid or fluid. A weak solu¬
tion of sulphate or phosphate of quinine would do very well, or a
weak decoction of the bark of the horse-chestnut (no doubt a
solution of pure esculine would be better), or an alcoholic solution
of the seeds of the Datura stramonium . But perhaps the most
convenient thing of all would be a slab of glass coloured yellow by
oxide of uranium. This would be always ready, and in point of
sensibility the glass does not seem to yield to any of the solutions
above mentioned, at least so far as relates to those rays which are
capable of passing through glass*

192. In making experiments on the invisible rays, it is well
to get rid, as far as possible, of the glare arising from the bright
part of the spectrum, and therefore a clear solid or solution is
preferable to an opaque screen. If it be desired to show the fixed
lines in the visible and invisible parts of the spectrum at the same
time, a screen may be employed consisting of paper washed with a
moderately strong solution of sulphate of quinine, or an alcoholic
solution of stramonium seeds. Turmeric paper is not, I think,
quite so good for showing the fixed lines of very high refrangi-
bility, but is at least equally good for the extreme violet and for
the rays a good distance further on, especially if if has been
washed with a solution of tartaric acid. If is likely that many
other acids would do as well. Very excellent screens might pro¬
bably be prepared by washing paper with a solution of esculine,
or even of the bark of the horse-chestnut*f*, or by covering paste¬
board with yellow uranite reduced to fine powder, and made to
adhere by a weak solution of pure gum Arabic; hut these I have
not tried.

Application of internal dispersion to demonstrating the course

of rays.

193. Solutions of quinine have already been employed for
this purpose, and a weak decoction of the bark of flic horse-chest¬
nut appears to be decidedly better. But the effect is immensely
improved by using absorbing media to cut off all the rays belong¬
ing to the bright part of the visible spectrum. A deep blue glass
will answer very well for this purpose if its faces be even, so as

See note F.

t See note G.

ON THE CHANGE OF REFRANGIBILITY OF LIGHT. 371

not to disturb the regularity of the refraction. The appearance of
the general pencil refracted through a rather large lens, with its
caustic surface, its geometrical focus, &c., is singularly beautiful
when exhibited in this way, on account of the perfect continuity
of the light, and the delicacy with which the different degrees of
illumination belonging to different parts of the pencil are repre¬
sented by the different degrees of brightness of the dispersed light.
The solution should be contained in a vessel with plane sides of
glass, and ought to be very weak, or else only the part of the
pencil which lies near the surface by which the light enters will
be properly represented.

Application of internal dispersion to the determination of the
absorbing power of media with respect to the invisible rays
beyond the violet , and the reflecting power of surfaces with
respect to those rays.

194. Hitherto no method has been known by which the ab¬
sorbing power of a medium with respect to these rays could be
determined for each degree of refrangibility in particular, except
that which consists in taking a photographic impression of a pure
spectrum, the light forming the spectrum having been transmitted
through the substance to be examined. It is needless to remark
how troublesome such a process is when contrasted with the mode
of determining the absorption which media exercise on the visible
rays. But the phenomenon of internal dispersion furnishes the
philosopher, so to speak, with eyes to see the invisible rays, so that
the absorbing power of the medium with respect to these rays
may be instantly observed. For this purpose it is sufficient to
form a pure spectrum, using instead of a screen a highly sensitive
fluid or solid, such as one of those mentioned in Art. 191, and to
hold before it the medium to be examined, or else to place the
medium over the whole or a part of the slit.

195. In this way the transparency of glass coloured yellow
by oxide of silver with respect to the violet rays and some of
those still more refrangible, which has been remarked by Sir John
Herschel*, may be at once observed. A set of green glasses were
found to be very variable in the mode in which they absorbed the

* Philosophical Transactions for 1840, p. 39.

24—2

372 ON THE CHANGE OF REFBANGIBIUTY OF LIGHT.

invisible rays, some absorbing the more refrangible of the rays
capable of affecting a dilute solution of sulphate of quinine and
transmitting the less refrangible, others absorbing the less and
transmitting the more refrangible, and others again absorbing
them all. These rays were absorbed by solutions of chromate and
bichromate of potash so weak as to be almost colourless. A thick¬
ness of about a quarter of an inch of sulphuret of carbon was suffi¬
cient to absorb all the rays beyond Hkl, so that a hollow prism
filled with this fluid would be useless in experiments on these rays.
It should be remarked that the sulphuret of carbon employed was
not yellow from dissolved sulphur, but apparently as colourless as
water.

196. To determine qualitatively the reflecting power of a
polished surface with respect to the invisible rays of each particu¬
lar degree of refrangibility, it would be sufficient to form a pure
spectrum as usual, reflect the rays sideways before they come to
the focus of the larger lens, place a sensitive medium to receive
them, and compare the effect with that produced on the same
medium when the rays are allowed to fall directly upon it.

Effect of different Flames.

197. Want of sunlight proved to be such an impediment to
the pursuit of these researches that I was induced to try some
bright flames, with the view of obtaining some convenient, substi¬
tute. Candle-light is very ill adapted to these experiments. The
flame of a camphene-lamp proved no better, perhaps rather worse,
for it abounds so much in rays belonging to the bright, part, of the
spectrum that the glare of the light prevents all observation of
faint objects; and the flame does not appear to he rich in invisible,
rays in anything like the proportion in which it is rich in visible
ones. The flame of nitre burning on wood or charcoal produced a
very good effect, exhibiting, when the combustion was most, vivid,
a copious dispersive reflexion in a weak solution of sulphate of
quinine contained in a bottle held near it. The tint of the dis¬
persed light appeared to be not quite the same as that given by
daylight, but to verge a little towards violet. However, I do not
place very strong reliance on the judgment of the eye under such
circumstances, A still stronger dispersive reflexion was produced

ON THE CHANGE OF REFRANGIBILITY OF LIGHT. 3*73

by a flash of gunpowder. The tint in this case appeared to be
the same as that seen by daylight.

198. While engaged in some of these experiments on bright
flames, I was surprised by discovering the strong effect produced
by the flame of a spirit-lamp, the illuminating power of which is
so feeble. When this flame was held close to a bottle containing
sulphate of quinine, a very distinct dispersive reflexion was
exhibited. The same was the case with several other sensitive
solutions. However, the full effect of the flame is not thus ex¬
hibited, because a considerable portion of the rays which it emits
is stopped by glass. It is best observed by pouring the solution
into an open vessel, such as a wine-glass or tumbler, holding the
flame immediately over it, and placing the eye in or very little
below the plane of the surface. In this way nothing is interposed
between the flame and the fluid, except an inch or two of air, the
absorption produced by which, it is presumed, is insensible; and
the plane strata, parallel to the surface, into which the illuminated
portion of the fluid may be conceived to be divided, are all pro¬
jected into lines, whereby the intensity of the blue light is materi¬
ally increased. I t is to be observed further, that if the eye be
held a little below the plane of the surface, there enters it, not
only the light coming directly from the blue stratum itself, but
also that coining from its image formed by total internal reflexion.
This mode of observation has already been employed by Sir John
Herschel in the case of sunlight. As it is frequently useful in
these researches it will be convenient to have a name for it, and I
shall accordingly speak of it as the method of observing by super¬
ficial projection.

199. The opacity of a solution of sulphate of quinine appears
to increase regularly and rapidly with the refrangibility of the
light. Hence we may form an estimate of the refrangibility of
any light by which the solution may be affected, by observing
the degree in which the illumination is concentrated in the
neighbourhood of the surface. For this purpose it is essential
to employ a weak solution, since otherwise streams of invisible
light of various degrees of refrangibility produce each their full
effect in strata so very narrow, that they cannot be distinguished
by the breadth of the stratum. Now to judge by the great con¬
centration of the illumination produced by a spirit-lamp, even in

374 ON THE CHANGE OF RE FRAN GIB I LIT Y OF LIGHT.

the case of an extremely weak solution, as well as by the con¬
siderable degree in which the active rays were intercepted by
glass, these rays, taken as a whole, must have been of very high
refrangibility, such as to place them among the most refrangible
of the fixed lines represented in the map, or perhaps even alto¬
gether beyond them. In making observations on the solar spec¬
trum, it was plain that the prisms were by no means transparent
with respect to the rays belonging to the group p of fixed lines.
Yet these rays, before they produced their effect, had to pass
twice through the plate-glass belonging to the mirror (except
so far as regards the rays reflected at the first surface), then
through three prisms, though to be sure as close as possible to
the edges, then through a lens by no means very thin, and
lastly, through the side of the vessel containing the fluid. Such
a train of glass would be sufficient materially to weaken, if not
even wholly to cut off the active rays coming from the flame of a
spirit-lamp.

200. The flame of naphtha* produces nearly 1 he same effect
as that of alcohol. The flame of ether is not so good ; but
whether this arises solely from its richness in visible rays, which
only produce a glare, or likewise from a comparative poverty in
highly refrangible invisible rays, it is not easy to say. The flame
of hydrogen produces a very strong effect. The invisible rays in
which it so much abounds, taken as a whole, appear to be even
more refrangible than those which come from t he flame of a
spirit-lamp. In making some observations with tin* flame of
hydrogen, when the gas was nearly exhausted, so that the flame,
was reduced to a roundish knob no larger than a, sweet pea, and
giving hardly any light, it was found still to produce a very
marked eflect when held over the surface of a solution of sul¬
phate of quinine. The flame of sulphuret of carbon produces on
most objects a much stronger effect than that of alcohol. It.
exhibits distinctly the blue light dispersed close, to the surface
of a solution of guaiacum in alcohol, which tin* flame of alcohol
does not. It appears then that the flame of sulphuret of carbon
is rich in invisible rays of such a refrangibility as to place them
among the groups of fixed lines m, n, or a little beyond, since

[* By this was meant wood-spirit, commercially called naphtha, not the
hydrocarbon to which the name more properly belongs.]

ON THE CHANGE OF REFRANGIBILITY OF LIGHT. 375

when a solution of guaiacum is examined in the solar spectrum,
it is found that that is the region in which the blue dispersed
light is produced. The blue light dispersed by a solution of
guaiacum may also be seen by using the blue flame of sulphur
burning feebly. The poverty of the flame of a spirit-lamp, not
only with respect to visible rays, but also with respect to in¬
visible rays, except those of very high refrangibility, accounts for
the circumstance that it does not exhibit, or at least hardly at all
exhibits, the blue light dispersed by fluor-spar.

Mode of determining , by means of the light of a spirit-lamp, the
transparency of bodies with respect to the invisible rays of
high refrangibility.

201. If the body be a solid, and be bounded by parallel
surfaces, its transparency with regard to these rays is easily
tested. For this purpose it is sufficient to hold the flame of a
spirit-lamp a little way above the surface of a weak solution
of sulphate of quinine contained in an open vessel in a dark
room, and then, placing the eye so as to see the dispersed light
in projection, alternately to interpose and remove the plate to be
examined.

202. On examining in this way various specimens of glass, I
found none which did not show evident defects of transparency.
The purest specimens of plate-glass appeared, I think, to be the
least defective. I cannot say whether the observed defects of
transparency were due to the essential ingredients of the glass,
or to accidental impurities. It is possible that glass made with
chemically pure materials might be transparent*. I believe that
a mere trace of peroxide of iron, or of sulphuret of soda or
potassa, would be sufficient to impair materially the transparency
of glass with respect to these rays, and such impurities are very
likely to be present. Quartz, however, appeared to be perfectly
transparent, the active rays passing through the thickness of one

* Some specimens of glass belonging to Dr Faraday’s experiments, which from
the absence of colour and of internal dispersion seemed hopeful, could not be
examined for transparency, on account of their irregular figure; and as they were
only lent to me by a friend, I did not feel myself at liberty to get them cut and
polished.

376 ON THE CHANGE OF REFRANGIBIUTY OF LIGHT.

or two inches, whether parallel or perpendicular to the axis, with¬
out any perceptible loss. The contrast between quartz and mica
was very striking, for a plate of mica no thicker than paper pro¬
duced a very sensible diminution in the illumination.

203. For the purpose of observing fluids, I procured two
vessels consisting of sections of a wide glass tube, about an inch
long, closed at one end with a disc of quartz. I shall call these
for brevity quartz vessels, though of course the bottom is the only
part in which there is any occasion to use quartz. When a fluid
is to be examined it is poured into a quartz vessel, and then the
vessel with its fluid contents is examined in the manner of a solid
plate, as described in Art. 201. On account of the perfect trans¬
parency of quartz, the fluid is as good as suspended in air. When
a quartz vessel was partly filled with water, the addition of a very
small quantity of nitrate of iron was sufficient to cause the ab¬
sorption of the active rays. The solution was so weak as to be
almost colourless when viewed through the thickness through
which the rays would have to pass. A solution of perehloride of
iron had a similar effect. These fluids I had specially examined
by sunlight, and had not found in them the least trace of in¬
ternal dispersion. When a fluid exhibits internal dispersion, it
is almost always very opaque with regard to rays of high ref’rangi-
bility, as is shown, without any special experiment, in the course
of the observations by which the internal dispersion is exhibited ;
but it by no means follows conversely, that when a fluid is very
opaque with regard to these rays, though nearly transparent with
regard to the visible rays, it exhibits the phenomenon of internal
dispersion.

204. I have little doubt that the solar spectrum would be.
prolonged, though to what extent I am unable to say, by using
a complete optical train in every member of which glass was
replaced by quartz. Such a train would he rat.Inn* (expensive, but.
would not involve any particular difficulty of execution. If solid
prisms of quartz were used, half of the incident light would be
lost, on account of the double refraction of the substance, unless
the prisms were cut in a particular manner, which however would
seem likely to involve some difficulties, both in the exertit ion and
in the observations. But hollow prisms holding fluids might be
employed, having the two faces across which the light has to

ON THE CHANGE OF REFRANGIBILITY OF LIGHT. 377

pass made of quartz plates. For a reason already mentioned,
sulphuret of carbon cannot be employed for filling the prisms',
and the dispersive power of water is very low, but there appears
to be no objection to the use of a solution of some colourless
metallic salt. At least saturated solutions of sulphate of zinc
and of acetate of lead, the only salts I have tried with this view,
showed no defects of transparency when examined in quartz
vessels by means of the flame of a spirit-lamp and a solution of
sulphate of quinine*.

Effect of Hydrochloric Acid, &c. on Solutions of Quinine . Optical
evidences of combination in other instances.

205. Sir John Herschel, in his interesting paper already so
often referred to, observes that it is only acid solutions of quinine
which exhibit the peculiar blue colour, and that among different
acids the muriatic seems least efficacious (page 145).

For my own part I have tried solutions of quinine (not di¬
sulphate) in dilute sulphuric, phosphoric, nitric, acetic, citric,
tartaric, oxalic, and hydrocyanic acids, and also in a solution of
alum. In all these cases the blue colour of the dispersed light
was plainly seen by ordinary daylight, especially when the fluid
was examined by superficial projection. It was not easy to say
which solution answered best, but I am inclined to think that in
which phosphoric acid was used.

200. But when quinine was dissolved in dilute hydrochloric
acid the blue colour was not exhibited, not even when the fluid
was held in the sunlight, and examined by superficial projection.
Certain theoretical views led me to regard this as an evidence
of a more intimate union between quinine and hydrochloric acid
than between quinine and the acids first mentioned, and to try
whether the addition of hydrochloric acid to the solutions men¬
tioned in the preceding paragraph would not destroy the blue
colour. On trial this proved to be actually the case, so that even
sulphuric acid is incapable of developing the blue colour in a
solution of quinine in hydrochloric acid.

See note H.

378 ON THE CHANGE OF REF RANCH BIX IT Y OF LIGHT.

207. That the quinine was not decomposed when the blue
colour due to sulphate of quinine was destroyed by hydrochloric
acid, but only differently combined, was shown by adding a
solution of carbonate of soda, which produced a white precipi¬
tate; and when this was collected on a filter, washed, and re¬
dissolved in dilute sulphuric acid, it exhibited the blue colour
as usual.

208. The addition of a solution of common salt, instead of
hydrochloric acid, to the solutions mentioned in Art. 205, like¬
wise destroyed the blue colour. In the case of sulphuric acid
this is only what might have been confidently anticipated ; but
we should not perhaps have expected that quinine in combina¬
tion with a weak acid, such as citric, would decompose hydro¬
chlorate of soda, giving rise to citrate of soda and hydrochlorate
of quinine; yet this appears to be the nature of the reartion.

209. It might perhaps be supposed that the sulphuric acid
was only partially expelled from sulphate of quinine by hydro¬
chloric acid, and that the salt in solution was really a sort of
double salt, in which the same base, quinine, was combined with
sulphuric and hydrochloric acids in atomic proportion. But if so,
it is probable, though not certain, that t he same salt would be
formed on adding hydrochloric acid to a solution of disulphate
of quinine, even though the quantity were not sutlieient to rum-
bine with the whole of the disulphate. On this supposition, if
hydrochloric acid were added by small quantities at a, time to a
solution of disulphate of quinine, the blue colour ought not to be
developed; and when acid enough had been added it ought to
be incapable of being developed by the addition of sulphuric
acid; whereas, if the whole of tin* sulphurie acid lx* expelled
by hydrochloric acid, the blue colour ought to lx* first developed,
by the conversion of a portion of the disulphate of quinine into
a sulphate, and then destroyed, on the addition of more acid, by
the conversion of the sulphate into a hydroehlorate. On trying
the experiment with a solution of disulphate of quinine in warm
water, it was found that the blue colour was actually first de¬
veloped and then destroyed.

210. A practical conclusion which seems to follow from these
results is, that in the employment of quinine in medicine it is of

ON THE CHANGE OF REFRANGIBILITY OF LIGHT. 379

little consequence whether the sulphate, phosphate, acetate, or
hydrochlorate be used, since the first three salts would be im¬
mediately converted by the common salt in the body into the
hydrochlorate, and the small quantity of a neutral salt of soda
resulting from the double decomposition could hardly, one would
suppose, be worth considering. However, the common quinine
is associated with cinchonine, the reactions of which may be
different. According to Sir John Herschel, the latter alkaloid
does not exhibit the blue colour, and therefore the optical tests
do not apply to it. If it be desired to obtain a soluble salt of
quinine which shall not be converted by common salt, by double
decomposition, into a hydrochlorate, it must apparently be sought
for among the combinations of quinine with very weak acids, the
affinity of which for soda does not much help that of hydrochloric
acid for quinine. It seems likely enough that such salts may
exist; for though acetate or citrate of quinine decomposes hydro-
chlorate of soda, hydrochlorate of quinine is decomposed by car¬
bonate of soda; and it is probable that many vegetable acids
behave like the carbonic in this respect.

211. The blue dispersion of a solution of sulphate of quinine
is destroyed by hydrobromic and hydriodic acids just as by hydro¬
chloric. In the experiment, solutions of bromide and iodide of
potassium wore used; but as a considerable excess of sulphuric
acid was purposely added to the solution of quinine, the potassa
introduced would merely remain inert in the solution as a sulphate,
without impeding the observation. The same experiment was
tried with phosphate of quinine with the same result.

212. It is stated in Turner’s Chemistry, that the play of
colours observed in solutions of polychrome (i.e. esculine) is de¬
stroyed by acids, and heightened by alkalies. The destruction, or
at least almost complete destruction, of the blue colour due to
dispersed light m a decoction of the bark of the horse-chestnut,
which is produced by acids, is readily observed; but I could not
perceive that the addition of alkalies in the first instance to a
fresh solution made any difference one way or other. If the blue
colour had previously been destroyed by an acid, it was restored
by the alkali. If the horse-chestnut had never been examined
chemically, these observations alone would indicate that in all

380 ON THE CHANGE OF KEFRANGIBXLITY OF LIGHT.

probability the principle to which the blue colour was due was
capable of entering into firm combination with acids, but did not
combine with alkalies. It is, in fact, as we know, a vegetable
base*

213. A solution of nitrate of uranium in ether is insensible,
as if some of the elements of the ether entered into firm combina¬
tion with the oxide of uranium. In connexion with this circum¬
stance, it is rather remarkable, that although the ether passes off
by evaporation when the solution is left to itself in an open
vessel, if heat be applied chemical action sets in, and the residue
consists chiefly of a salt which has all the appearance of oxalate
of uranium. This salt, when washed and examined in the moist
state, without very great concentration of light, was found to be
insensible f.

214. It is rare to meet with solutions so highly sensitive as
those of quinine and esculine, but similar observations may be
made on a great number of solutions, by employing suitable
methods. The most searching method consists in forming a
bright and tolerably pure spectrum, by transmitting the sun’s
light through a very broad slit, or even leaving out the slit
altogether. It is desirable to use a lens of only moderate foral
length in connexion with the prisms. The solution having been
placed in the spectrum, the acid, or other agent whose reactions
it is desired to study, is to be added, and tin* effect, if any,
observed. It is usually advantageous to cover the slit with a, blue
glass, or similar absorbing medium; but sometimes effects take
place in the bright part of the spectrum, which is intercepted by
such a medium. When false dispersion abounds, it, is well to look
down on the fluid through a Nmols prism, so as to stop all light
which is polarized in the plane of reflexion.

Negative results with reference to a mutual action of the rays
incident on sensitive solutions.

215. The antagonistic effects of the more and less refrangible
rays, which have been observed in certain phenomena, induced

* [It is not a base but a glueoside.J
+ See note I.

ON THE CHANGE OF REFRANGIBILITY OF LIGHT. 381

me to try whether anything of the kind could be perceived in the
case of internal dispersion. The following arrangement was
adopted for putting this question to the test of experiment.

A tumbler was filled with a very dilute solution of sulphate of
quinine, and placed in a pure spectrum. As usual, the illuminated
portion of the fluid consisted of two distinct parts, one the blue
beam of truly dispersed light, corresponding to the highly re¬
frangible rays, the other the beam reflected from motes, exhibiting
the usual prismatic colours, and corresponding to the brighter of
the visible rays. The fluid was nearly free from motes, so that
the first beam was by far the brighter of the two; and the second
beam, without being bright enough at all to interfere with the
observation, was useful as serving to point out where the red,
yellow, &c. rays lay. A flat prism, having an angle of about 130°,
was then held in front of the vessel, with its edge vertical, and
situated in the more refrangible part of the visible rays. The
rays forming the two beams were thus bent in opposite directions,
and the beams made to cross each other within the fluid; and by
turning the prism a little in both directions in azimuth, that is,
round an axis parallel to the incident rays, it was easy to make
sure that, the beams did actually cross. But not the slightest
perceptible difference in the blue beam was made by the passage
of the r<id and other lowly refrangible rays across it.

21(j. Certain theoretical views having led me to regard it as
doubtful whether the intensity of light internally dispersed was
proportional to the intensity of the incident rays, other circum¬
stances being the same, I was induced to try the following experi¬
ment.

The sun’s light was reflected horizontally through a large lens,
which was covered by a. screen containing two moderately large
round holes, situated in the same horizontal plane, and a good
distance apart. The beams coming through the two holes con¬
verged of course towards the focus of the lens, and at the same
time contracted in width, and became brighter from the concen¬
tration of the light. For our present purpose, they may be
regarded as cylindrical beams converging towards the focus of
the Ions. When they had approached each other sufficiently,
they were transmitted through a blue ammoniacal solution of

382

ON THE CHANGE OF REFRANGIBILITY OF HOIIT.

copper, contained in a vessel with parallel sides. The object of
this was of course to absorb all the bright visible rays, which
would not only be useless for exciting the solution which it was
meant to try, but would materially hinder the observation by the
glare which they would produce. The beams were then admitted
into a vessel containing a decoction of the bark of the horse-
chestnut, greatly diluted with water. In passing through the
fluid they produced two blue beams of truly dispersed light, which
converged towards a point a little way outside the vessel. A flat
prism, with an angle of about 150\ was then held in front of the
vessel, with its edge vertical, and situated between the incident
beams. The blue beams of dispersed light were thus made to
cross within the fluid; and by moving the prism in azimuth, it
was easy to make one beam either fall above the other, cross it,
or fall below it. Now on looking down from above with one eye
only, and moving the prism backwards and forwards in azimuth,
I could not perceive the slightest difference of illumination,
according as the blue beams actually crossed each other, or were
merely seen projected one on the other. In this experiment,
then, it appeared that one beam of incident rays produced as
much additional dispersed light in a portion of fluid already
excited by the other beam, as it was capable of producing in a
similar portion of fluid not otherwise, excited.

Effect of an electric spark. Nature of its phosphorotjeuic rat/s.

217. For the use of the apparatus with which tin* following
experiments were made, I am indebted to the, kindness of Professor
Cumming.

An electric spark produces an internal dispersion of light in a
very striking manner in the case of an extremely dilute solution
of sulphate of quinine. Having prepared a solution so weak,
that when it was examined by superfieial projection by the light*
of a spirit-lamp, nothing was semi but a pale gleam of light
.extending a good way into the fluid, and not only not confined to
the surface, but not even showing any particular concentrat ion
in the neighbourhood of the surface, 1 placed it so as to be
illuminated by the sparks from the prime conductor of an elcctri-
flying machine, which passed at no great distance over the surface.

ON THK CHANGE OF REFRANGIBILITY OF LIGHT. 383

A very marked internal dispersion was produced, but the nature
of the effect depended in a good measure on the character of the
spark. A feeble branched spark, giving but little light, and
making little noise, produced an illumination extending to a
considerable depth, and very much stronger than that occasioned
in the same solution by the flame of a spirit-lamp. The rays by
which this was produced passed in a great measure through a
plate of glass interposed between the spark and the surface of the
fluid. But a bright linear spark, making a sharp crack, produced
an illumination almost confined to an excessively thin stratum
adjacent to the surface of the fluid; and the rays by which this
was produced were cut off by glass, though transmitted through
quartz. The same was the case with the discharge from a Leyden
jar, which produced a bright light almost confined to the surface*.

218. The opacity of a solution of sulphate of quinine appears
to increase regularly and rapidly with the refrangibility of the
rays incident, upon it. Hence we are led to the conclusion that a
strong electric spark is excessively rich in invisible rays of ex¬
tremely high refrangibility. Gliiss is opaque with respect to these
rays, but. quartz transparent.

2IS). It is known that the phosphorogenic rays of an electric
spark, at least those which affect Canton’s phosphorus, pass very
freely through quart, z, but are stopped by a very moderate thickness
of glass. This alone, after what has been already mentioned,
would lead us to suppose that the phosphorogenic rays coming
from such a spark are merely rays of very high refrangibility. If
so, they ought, to he intercepted by a very small quantity of a
substance known to absorb such rays with energy.

After having made some experiments on the production of
phosphorescence in Canton’s phosphorus by means of an electric
discharge, and observed how the influence of the discharge was
transmitted through quartz and stopped, or almost entirely stopped,
by glass, I felt, confident that my own observations were com¬
parable with those of others. A small portion of the phosphorus
was then placed on card, covered by an empty quartz vessel, and
had the discharge of a Leyden jar passed over it. The phos-

See note J.

384 ON THE CHANGE OF REFRANGIBILITY OF LIGHT.

phorescence was powerfully excited, being visible in a room which
was by no means quite dark; and when the card was carried into
a dark place, the phosphorescent light remained plainly visible for
a good while. The experiment was then repeated with a fresh
portion of the same phosphorus, the vessel this time containing
water. The phosphorescence was produced as before, though not
I think so copiously. But on taking a fresh portion of the
phosphorus, and substituting for water a very dilute solution of
sulphate of quinine, the influence of the spark was arrested, and
the phosphorus was not rendered luminous. It was found that a
solution containing only about one part of quinine in 10,000, with
a depth of half an inch, was sufficient to prevent the generation of
phosphorescence.

220. This result, it seems to me, would be sufficient, were
proof wanting, to show that no part of the effect is attributable
directly to the electrical disturbance. The effect produced when
the phosphorus is at the distance of an inch or so from the points
of the discharger seems exactly the same as when it is nearer,
being merely somewhat weaker, as would naturally be expected,
whatever view were taken of the nature of the influence. But at
the distance of an inch, the influence of the spark, though it
passes freely through quartz and water, is cut off by adding to the
water an excessively small quantity of sulphate of quinine. It
cannot be supposed that the electrical relations of the. medium, or
its permeability to electrical attractions and repulsions, are utterly
changed by such an addition; while, on the other hand, tho result
is in perfect conformity with what we know respecting the stop¬
page of radiations by absorbing media. However, the principal
object of the experiment was not to confirm the view which
makes the influence of the spark to consist in tho rays which
emanate from it, a view which I suppose is pretty generally
adopted, but to investigate more fully the nature of these rays.
Enough has, I think, been adduced to show that they are merely
rays which there is no reason to suppose are physically different
from those of light, but quite the contrary, and which are of very
high refrangibility, and are therefore invisible, since they fall far
beyond the limits of refrangibility within which the retina is
affected. Indeed, it seems very likely that the highly refrangible
rays never reach the retina, but are absorbed by the coats of the

ON THE CHANGE OF RE FRAN OIBILITY OF LIGHT.

385

eye*. Hence the phenomena relating to the phosphorescence
produced by an electric discharge afford no countenance to the
supposition that it is possible to divide rays of a given refrangi-
bility into phosphorogenic, chemical, luminous, &c. Of course the
most unexceptionable mode of determining the refrangibility of
the phosphorogenic rays would be by actual prismatic decom¬
position, but this would require the employment of a quartz
train.

Points of resemblance and contrast between internal dispersion
and phosphorescence.

221. As the term phosphorescence has been applied to several
different phenomena, I must here explain that I mean the sponta¬
neous exhibition of a soft light, independently of chemical changes,
which some substances exhibit for a time after having been
exposed to the sun's rays, or to an electric discharge, or to light
from some other sources.

In many respects the two phenomena have a strong resem¬
blance. Thus, the general features of internal dispersion cannot
be better conceived than by regarding the sensitive medium as
self-luminous while under the excitement of the active rays.
Again, it is well known that the rays of the solar spectrum by
which the phosphorescence of Canton’s phosphorus, sulphuret of
barium, and other phosphori, is produced, are those of high
refrangibility, as well as the invisible rays beyond; and these are
precisely the rays which in the great majority of cases are most
efficient in producing internal dispersion. I do not however know
how far it may bo true that when phosphorescence is excited by
homogeneous light the refrangibility of the incident light is a
superior limit to the refrangibilities of the component parts of the
light, omitted. Indeed, according to Professor Draper, when the
phosphorescence of Cantons phosphorus is excited by the rays
from incandescent lime, the active rays belong to the red extremity
of the spectram*|\ If this result be confirmed^, it follows that the

* Sec note K.

t Philosophical Matjazine, Vol. xxvit. (Dec. 1845) p. 436.

X [Early in 1853 1 wan engaged along with Faraday in preparing in the labora¬
tory of the Royal InHtitution some experiments to show at an evening lecture on
the subject of this paper. I expressed to him a wish to repeat Draper’s experiment,

25

s. III.

386 ON THE CHANGE OF REFRANGIBILITY OF LIGHT.

most striking law relating to internal dispersion is not obeyed in
the case of phosphorescence.

In the same paper Professor Draper remarks, “Some time ago
I determined the refrangibility of the rays of an electric spark
which excite phosphorescence in sulphnret of lime ; they are found
at the violet extremity of the spectrum ” In what way Professor
Draper determined the refrangibility of rays with respect to which
glass is so opaque, he does not give the least hint. Being perfectly
in the dark as to the evidence on which the conclusion is based, I
cannot accept it in contradiction to my own experiments. Perhaps,
however, “at the violet extremity” may mean nothing more than
somewhere in the highly refracted region beyond the visible rays.
If so, Professor Drapers statement is in accordance with my own
conclusions.

222. When one part of a phosphorus has been excited, the
phosphorescence is found gradually to extend itself to the neigh¬
bouring parts*. In this respect a substance which exhibits internal
dispersion presents a striking contrast. The finest fixed lines of
the spectrum are seen sharply defined, whether in a solution, or in
a clear solid, or on a washed paper.

223. Of course, theoretically, there ought, to a certain extent,
to be a communication of illumination from one part of a sensitive
fluid to another, on account of the light which is twice, three

as it had such an important bearing on the subject, and he immediately tried it.
We obtained, however, only a negative result, as the Canton’s phosphorus which
had been acted on by bright light which had been passed through a solution of
bichromate of potash did not give out in the dark any sensible light. Long
afterwards, as I was engaged with some experiments on Balmain’s luminous paint,
it occurred to me that possibly Draper’s result might have been due to a latent
effect of a previous exposure to light; and I wrote to him to enquire whether this
might have been possible. He replied that he generally heated his phosphori
before proceeding to an experiment, in order to guard against the possible existence
of a latent effect of previous exposure ; that at that distance of time he could not
be certain whether that had been done in the particular experiment referred to;
that if it had not, the result he had mentioned might have been brought about in
that way.]

* [M. Becquerel afterwards explained this apparent result by attributing it to
the increased sensitiveness of the eye arising from continuance in the dark, which
enabled the observer to see outlying portions of a phosphorescing patch which
were not seen at first, thereby giving the impression that the phosphorescence was
extending.]

ON THE CHANGE OF REFRANGIBILITY OF LIGHT. 387

times, &c., dispersed. This however must be excessively small;
for the mean refrangibility of the dispersed light is usually much
lower than the refrangibility of the active light, perhaps lower
than that of any light capable of exciting the solution. However,
generally some few of the dispersed rays would have a refrangi¬
bility sufficiently high to be dispersed again. But practically the
intensity of the light twice dispersed in this manner would be so
very small that it may safely be altogether disregarded.

224. But by far the most striking point of contrast between
the two phenomena, consists in the apparently instantaneous com¬
mencement and cessation of the illumination, in the case of internal
dispersion, when the active light is admitted and cut off. There
is nothing to create the least suspicion of any appreciable duration
in the effect. When internal dispersion is exhibited by means of
an electric spark, it appears no less momentary than the illumina¬
tion of a landscape by a flash of lightning. I have not attempted
to determine whether any appreciable duration could be made out
by means of a revolving mirror *.

225. There appears to be no relation between the substances
which exhibit a change of refrangibility and those which phos¬
phoresce, either spontaneously, or on the application of heat.
Thus the sulphurots of calcium and barium, on being examined
for internal dispersion, were found to be insensible, as was also
Iceland spar. The last substance phosphoresced strongly on the
application of heat. So far as was examined, the minerals which
did exhibit, a change of refrangibility showed no special disposition
to phosphoresce. Sir David Brewster has remarked, that a speci¬
men of fluor-spar which exhibited a blue light by internal disper-

* [The experiment I had in my mind was to view in a revolving mirror the
substance to be examined while illuminated in a dark room by a succession of
sparks from the prime conductor of an electrifying machine, taking one’s chance
for the casual appearance of images in the field of view. The experiment was
afterwards tried with apparatus kindly lent me, but whether with sparks from a
prime conductor or with an induction coil I am not now sure. Notwithstanding
what M. JBecquerel had in the mean time done with his beautiful phosphoroscope,
the results obtained are not perhaps wholly without interest. Thus yellow
uranito instead of its usual appearance showed a well-defined image of a very
ordinary looking yellow stone and a long drawn out gleam of green light due of
course to the fading phosphorescence.]

25—2

388 ON THE CHANGE OF RKFRANGIBILITY OF LIGHT.

sion, exhibited when heated a blue phosphorescent light; but this
appears to have been merely a casual coincidence *.

On the Cause of True Internal Dispersion , and of Absorption.

226. In considering the cause of internal dispersion, we may
I think at once discard all supposition of reflexions and refractions
of the vibrations of the luminiferous ether among the ultimate
molecules of bodies. It seems to be quite contrary to dynamical
principles to suppose that any such causes should be adequate to
account for the production of vibrations of one period from vibra¬
tions of another.

All believers, 1 suppose, in the undulatory theory of light are
agreed in regarding the production of light in the first instance as
due to vibratory movements among the ultimate* molecules of the
self-luminous body. Now in the phenomenon of internal disper¬
sion, the sensitive body, so long as it is under tin* influence of the
active light, behaves as if it were self-luminous. Nothing then
seems more natural than to suppose that tin* incident vibrations
of the luminiferous ether product* vibratory movements among
the ultimate molecules of sensitive substances, and that, tin* mole¬
cules in turn, swinging on their own account, product* vibrations
in the luminiferous ether, and thus cause the* sensation of light.
The periodic times of these vibrations depend upon tin* periods in
which the molecules are disposed to swing, not upon tin* periodic
time of the incident vibrations.

227. But in the very outset of this theory an objection will
probably be urged, that it is quite as much contrary to dynamical
principles to suppose the periodic time of tin* ethereal vibrations
capable of being changed through tin* intervention of ponderable
molecules as without any such machinery*!-. Tin* answer to this
objection is, that such a notion depends altogether on tin* applica¬
bility of a certain dynamical principle relating to indefinitely small
motions, and that we have no right to regard the molecular vibra-

* Report of the Meeting of the British Association at Newcastle in IHT.I, p. 11.

t [The attempt hero made to account for the lowering of refrangibility has long
since been given up, and I have been led to adopt what seems to me a far more
probable explanation. See Addition to this paper, p. 410.]

ON THE CHANGE OF RE F KANG IB I LIT Y OF LIGHT. 389

tions as indefinitely small. The excursions of the atoms may be,
and doubtless are, excessively small compared with the length of a
wave of light; but it by no means follows that they are excessively
small compared with the linear dimensions of a complex molecule.
It is vvell known that chemical changes take place under the
influence of light, especially the more refrangible rays, which
would not otherwise happen. In such cases it is plain that the
molecular disturbances must not be regarded as indefinitely small.
But vibrations may very well take place which do not go to the
length of complete disruption, and yet which ought by no means
to be regarded as indefinitely small. Furthermore, it is to be
observed that if in the cases of indefinitely small molecular dis¬
placements the forces of restitution be not proportional to the
displacements, the principle above alluded to will not be applicable
however small the disturbance may be; and if in tbe expressions
for the forces of restitution the terms depending on first powers of
the displacements (supposed finite), though not absolutely null,
be very small, the principle will not apply unless the molecular
excursions be extremely small indeed. In consequence of the
necessity of introducing forces not proportional to the displace¬
ments, it would be very difficult to calculate the motion, even
were we acquainted with all the circumstances of the case, whereas
we are quite in tin? dark respecting the actual data of the problem.
But certainly wo cannot affirm that in the disturbance communi¬
cated back again to the luminiferous ether none but periodic
vibrations would be produced, having the same period as the inci¬
dent vibrations. Rather, it seems evident that a sort of irregular
motion must be produced in the molecules, periodic only in the
sense, that the, molecules retain the same mean state; and that
the disturbance which the molecules in turn communicate to the
ether must be such as cannot be expressed by circular functions
of a given period, namely, that of the incident vibrations.

228. It is very remarkable with what pertinacity a particular
mode: of infernal dispersion attaches itself to a particular chemical
substance. Tims the, singular dispersion of a red light exhibited
by the green colouring matter of leaves is found in a green leaf, or
in a solution of the green colouring matter in alcohol, ether, sul-
phuret of carbon, or muriatic acid. The dispersion exhibited by
nitrate of uranium is found in a solution of the salt in water, as

390 ON THE CHANGE OF REFRANUIBIUTY OF LIGHT.

well as in the crystals themselves, which care doubly refracting.
In all probability therefore the molecular vibrations bv which the
dispersed light is produced are not vibrations in which the mole¬
cules move among one another, but vibrations among the con¬
stituent parts of the molecules themselves, performed by virtue of
the internal forces which hold the parts of the molecules together.
It is worthy of remark that it is chiefly among organic compounds,
the ultimate molecules of which we are taught by chemistry to
regard as having a complicated structure, that internal dispersion
is found. It is true that peroxide of uranium furnishes many
examples of internal dispersion ; but then the anhydrous peroxide
is itself insensible, it is only some of the compounds into which it
enters that are so remarkably sensitive ; and the chemical formulas
of these compounds, so far as they are known, are not by any
means extremely simple, although it is true that they may not he
more complicated than formuhe relating to other oxides. Why
this particular oxide should be disposed to enter into tottering
combinations I do not pretend even to conjecture; but it seems
not a little remarkable that peroxide of uranium, which is so
peculiar with respect to its optical properties, should also present
some singularities in its mode of chemical combination, which led
M. Peligot to regard it as the protoxide of a compound radical.

229. We are, I conceive, at present, far from an explanation
of the phenomena of internal dispersion in all their details. They
appear to be associated with the inmost, structure of chemical
molecules, to such a degree as to throw even tin* phenomena of
polarization into the shade. In this respect, indeed, absorption
seems superior to polarization, since most, of the phenomena of
polarization refer rather to the state of crystalline aggregation of
the molecules than to their constitution j but the phenomena of
internal dispersion appear to be much more searching than those
of absorption. There is one law however relating to internal
dispersion so striking and so simple, that it .swims not unreasonable
to look for an explanation ol it* I allude to that according to
which the refrangibility of light is always lowered in the process
of dispei'sion. I have not hitherto been able altogether to satisfy
myself respecting a dynamical explanation of this law, but the
following conjectures will not perhaps be deemed altogether
unworthy of being mentioned.

ON THK OIIANGK OF RKFIiANUIBIUTY OF LIGHT. 391

230. Reasons have already been brought forward for regard-
ing the molecular vibrations as performed under the influence of
forces not proportional to the displacements. For simplicity’s
sake, let us suppose for the present the parts of the forces of
restitution depending upon first powers of the displacements to
be absolutely null. Then, when a molecule is disturbed, its atoms
will be acted on by forces depending upon the second and higher
powers of the displacements. These forces must tend to restore
the atoms to their mean positions; otherwise the equilibrium
would be unstable, and the atoms would enter into new combina¬
tions, either with one another, or with the atoms of the surround¬
ing medium; so that, in fact, such compounds could never be
formed. The condition of stability would require the parts of the
forces depending upon squares of the displacements to vanish, but
this is a point which need not be attended to, all that is essential
to bear in mind being, that we have forces of restitution varying
in a higher ratio than the displacements. If the parts of the
forces of restitution which depend upon first powers of the dis¬
placements, though not absolutely null, be very small, the remain¬
ing parts must still be such as to tend to restore the atoms to
their positions of equilibrium ; otherwise the stability of the mole¬
cule, though not. mathematically null, would be so very slight, that
such compounds would probably never form themselves, but others
of more stability would be formed instead. Or, even were such
unstable compounds formed, they would probably be decomposed
on attempting to excite them in the manner in which sensitive
substances are excited in observing the phenomena of internal
dispersion; so that whether they exist or not, they may be set
aside in considering these phenomena.

2:\ 1. Now wlum vibrations arc performed under the action of
forces which vary in a higher ratio than the displacements, the
periodic times are not constant, but depend upon the amplitudes
of vibration, being greater or less according as. the amplitudes
are less or greater. Suppose the molecular and ethereal vibrations
already going on, and imagine the amplitudes of the former kept
constant by the application of external forces. According to the
value of the epoch of the vibrations of a particular molecule,
the ethereal vibrations will tend, in the mean of several successive
undulations, to augment or to check the vibrations of the molecule.

392 ON THE CHANGE OF REFRANGIBIUTY OF LIGHT.

For some time there will be a tendency one way, them for Home
time a tendency the other way, and so on, the opposite tendencies
balancing each other in the long run. The lengths of the times
during which the tendency lies in one direction, will depend upon
the periodic times of the molecular and ethereal vibrations, being
on the whole greater or less according as the two periodic times
are more or less nearly equal But since no external forces
actually act to keep the amplitudes constant, when the ethereal
vibrations are favourable to disturbance the molecule is further
disturbed, and therefore its periodic time is diminished; and
when they are favourable to quiescence the disturbance of the
molecule is checked, and therefore its periodic time is increased.
If, then, the ether be vibrating more rapidly than the molecule,
when the action is favourable to disturbance the periodic time of
the molecular vibrations is rendered more nearly equal to that
of the ethereal vibrations, and therefore the time during which
the action is favourable to disturbance is prolonged ; but when
the action is favourable to quiescence, tin* effect is just the
reverse. Hence, on the whole, there is a balance outstanding in
favour of disturbance. But if the ether he vibrating more slowly
than the molecule, it appears from similar reasoning that there
will be a balance the other way. Hence it is only when the
periodic time of the ethereal vibrations is less than that of file
molecular, that the latter vibrations can be kept going by the
former.

232. But it will probably be objected to this explanation,
that when a periodic disturbing force affects the moan motion of
a planet, the mean motion is a maximum, not when the force
tending to augment it is a maximum, but at a time later by a
quarter of the period of the force, namely, when the, force vanishes
in changing sign; and that in a similar manner the change* in
the periodic time of the vibrations of a disturbed molecule will
affect equally the duration of the time during which tin* act ion
is favourable to increased disturbance, and that, during which it. is
favourable to quiescence, or more exactly will not alter either,
since the effects in the first and second halves of those times will
neutralize each other. The answer to this objection is, that we
must not treat a molecule as if it were isolated, like a heavenly
body, since it is continually losing its motion by communication,

ON THE CHANGE OF RKFHAN(JIHUJTV OF LIGHT.

393

perhaps to neighbouring molecules, but at any rate to the lumi¬
niferous other; for without, a communication of the latter kind
there would be no dispersed light. Hence we must consider the
immediate tendency of (he disturbing forces rather than their
tendency in tin 1 long run.

233. When a molecule itself vibrates in an irregularly
periodical manner, the vibrations which it imparts to the ether
are of course of a similar character. The resolution of these
into vibrations corresponding to different degrees of refrangibility
involves some very delieate mathematical considerations, into
which I do not propose to enter. But without this it is evident
that when the other is agitated by the vibrations of an immense
number of molecules, in all possible states as regards amplitude,
ami consequently periodic time of vibration, the disturbance of
the ether must consist of a mixture of periodic vibrations, having
their periods comprised between the greatest and least of those
belonging to tin* molecular vibrations; and corresponding to those
different periods there will be, portions of light of different degrees
of refrangibility found in the dispersed beam. These refrangi-
bilities will range between two limits, an inferior limit equal to
the refrangibility corresponding to the periodic time of indefinitely
small vibrations, and a superior limit equal to the refrangibility of
the active light*.

234. This theory seems to accord very well with the general
character of dispersed beams, as regards the prismatic composition
of the light of which they consist. When analysed by a prism,
these beams a,re, sometimes found to break off abruptly at their
more refrangible border, but I do not recollect ever to have met
with an instance in which a beam broke off abruptly at the
opposite border, except when the whole beam was almost homo¬
geneous. This is just as it ought to be according to the above
theory, because the amplitude of vibration decreases indefinitely
in approaching the less refrangible limit. In the case of a
solution of chlorophyll, we may suppose that the part of the
molecular forces of restitution depending on first powers of the
displacements is considerable, on which supposition, the effect
ought to approach to what would take place were there no other
part. But were the forces of restitution strictly proportional to

394 ON THE CHANGE OK REKRANGIBIEITY OK EIGHT.

the displacements, the vibrations would be isochronous, and could
only he excited by ethereal vibrations having almost, exactly the
same period, but would be powerfully excited by such. Ac¬
cordingly, in a solution of chlorophyll the dispersion comes on
very suddenly; a large part of it is produced by active light
of nearly the same refrangibility as the dispersed light; and the
latter, by whatever active light produced, has nearly the same
refrangibility that it had at first. This supposition, combined with
the preceding theory, accounts also for the transparency of the
fluid with respect to rays of less refrangibility than the first
absorption band, for the great intensity of that band, for the
rapidity with which opacity comes on at its less refrangible
border, and the comparatively slow resumption of transparency
on the other side. A difference of the same nature on opposite
sides of a maximum of opacity seems to be a very common
phenomenon in absorption. On the other hand, in those numerous
cases in which the dispersion comes on gradually, in the manner
described in Art. 44, wc may suppose the part of the forces of
restitution depending on first powers of the displaeoments to he
hut small.

235. It may appear at first sight to be. a formidable, objection
to the theory here brought forward, that, in the experiment,
mentioned in Art. 21fi, tin; intensity of the dispersed light, did
not appear to be more than doubled when tin- intensity of the
incident disturbance was doubled; and that, in the experiment,
described in Art. 215, the rays of low refrangibility did not,
appear to exercise any protecting influence. Hut the difficulty
may, I think, be got over by a very reasonable supposition. It.
seems very natural to suppose that a given molecule remains for
the greater part of the time at rest,, or nearly so, and only now
and then gets involved in vibrations. On this supposition, it. is
only a very small per-contage of the molecules that, at a given
instant are vibrating to an extent worth considering. < lone.eive
now a'stream of light consisting of the highly refrangible rays to
be incident on a sensitive medium, and to cause. I per rent, of the
sensitive molecules to vibrate considerably, the rest vibrating so
little that they may he regarded as at rest. Now imagine a
second stream, similar in all respects to the first, to influence the
medium which is already under the influence of the first stream.

ON THE CHANGE OF REFRANGIBILITY OF LIGHT.

395

Of the 1 per cent, of the molecules already vibrating, many are
vibrating, we may suppose, nearly with their maximum amplitude,
and consequently are not much affected. Besides, it is a great
chance if the epoch of the ethereal vibrations belonging to the
second stream is such as to produce any great tendency either
towards quiescence or towards disturbance in a molecule just for
the short time that it is vibrating strongly under the influence of
the first stream. But of the 99 per cent, of quiescent molecules
1 per cent, are made to vibrate. Hence the effect of the two
streams together is very nearly the same in kind as that of one
alone, but double in intensity.

236. The apparent absence of a protecting influence in the
less refrangible rays seems at first more difficult to account for,
but perhaps the following reasoning may be thought satisfactory.
We ought not to attribute more influence in the direction of
protection to a second beam of rays of low refrangibility, than in
the contrary direction to a second beam of rays of high refrangi¬
bility. Now if the effect of a beam of rays of high refrangibility
be to throw 1 per cent, of the molecules into a state of vibration,
it would be a commensurate effect in a beam of rays of low
refrangibility to stop the vibrations of 1 per cent, of the molecules,
if they were all vibrating’ But since only 1 per cent, are actually
vibrating, the real protecting effect amounts to no more than
stopping the vibrations of one molecule in every 10,000, an effect
which may be regarded as insensible.

237. The simple consideration that work cannot be done
without the expenditure of power, shows that when light incident
on a medium gives rise to dispersed light, a portion at least of
the absorption which the medium is observed to exercise must be
due to the production of the dispersed light. If the dispersed
light really arises from molecular disturbances, and for my own
part I think it almost beyond a question that it does, it follows
that in these cases light is absorbed in consequence of its being
used up in producing molecular disturbances. But since we must
not needlessly multiply the causes of natural phenomena, we are
led to attribute the absorption of light in all cases to the pro¬
duction or augmentation of molecular disturbances, unless reason
be shown to the contrary. It might seem at first sight that the

396 ON THE CHANGE OF REFRANGIBILITY OF LIGHT.

production or non-production of dispersed light establishes at
once a broad distinction between different kinds of absorption.
I do not think that much stress can be laid on this distinction.
In the first place it may be remarked, that we have no reason to
suppose that vibrations which are of the same nature as those of
light are confined to the range of refrangibility that the human
eye can take in. If, therefore, no dispersed light be perceived,
it does not follow that no invisible rays are dispersed. If the
incident light belong to the visible part of the spectrum, the
dispersed rays (if any), being of lower refrangibility than the
incident light, can only be invisible by having a refrangibility
less than that of red light, and would manifest themselves solely
or mainly by their heating effect. However, though invisible rays
of this nature are in all probability emitted by the body in
consequence of the absorption of visible light, we are not bound
to suppose that in their mode of emission they precisely resemble
the visible rays observed in the phenomena of internal dispersion.
In most cases, perhaps, they are more nearly analogous to the
visible rays emitted by solar phosphori. It is possible to conceive,
and it seems probable that there exist, various degrees of molecular
connexion from mere casual juxtaposition to the closest chemical
union. A compound molecule may vibrate as a whole, by virtue
of its connexion with adjacent molecules, or it may vibrate by
itself, in the manner of an isolated vibrating plate or rod, and
between these extreme limits we may conceive various inter¬
mediate modes of vibration. Hence, without departing from the
general supposition that the absorption of light is due to the
production of molecular disturbances, we may conceive that the
modes in which the ether communicates its vibrations to the
molecules, and the molecules in turn communicate their dis¬
turbances to the ether, are very various.

I do not bring forward the idea that the absorption of light is
due to the production of molecular disturbances as new, though
possibly the communication of the ethereal vibrations to the
molecules may hitherto have been supposed necessarily to imply
the existence of synchronous vibrations among the molecules.
The change in the periodic time of vibrations which takes place
in the process of internal dispersion would hardly have been
suspected, had it not been for the singular phenomenon which
pointed it out.

ON THE CHANGE OF REFRANGIBILITY OF LIGHT. 397

238. The only theory of absorption, so far as I am aware, in
which an attempt is made to deduce its laws from a physical
cause is that of the Baron Von Wrede, who attributes absorption
to interference*. The Baron’s paper is in many respects very
beautiful, but it has always appeared to me to be a fatal objection
to his theory that it supposes vibrations to be annihilated. It is
true that two streams of light may interfere and produce darkness
but then to make up for it more light is produced in other
quarters. Light is not lost by interference, but only the illumina¬
tion differently distributed. Were the disappearance of light in
the direction of a pencil admitted into a medium merely a
phenomenon of interference, the full quantity of light admitted
ought to be forthcoming in side directions. W T ere a series of
vibrations incident on a medium, without producing any pro¬
gressive change in its state, or any disturbance issuing from it, it
would follow that work was continually being annihilated. But
we have reason to think that the annihilation of work is no less a
physical impossibility than its creation, that is, than perpetual
motion.

List of highly sensitive substances.

239. For the sake of any one who may wish to make experi¬
ments in this subject, I subjoin a list of the more remarkable of
the substances which have fallen under my notice. It will be
seen that most of these substances were suggested by the papers
of Sir David Brewster and Sir John Herschel.

Glass coloured by peroxide of uranium: yellow uranite: nitrate
or acetate of the peroxide. Probably various other salts of the
peroxide would do as well. The absorption bands of the salts,
whether sensitive or not, of peroxide of uranium ought to be
studied in connexion with the change of refrangibility.

A solution of the green colouring matter of leaves in alcoh^ 1
To obtain a solution which will keep, it is well previously to
the leaves in boiling water. The alcohol should not \
permanently in contact with the leaves, unless it be wisi
observe the changes which in that case take place, but por

* Poggendorff h Aimed cii, 33. xxxm. S. 353 ; or Taylor s Scientific Ifoi
Vol. i. p. 477.

398 ON THE CHANGfc OF REFRANGIBILITY OF LIGHT.

when the strength of the solution is thought sufficient. Also, the
solution when out of use must be kept in the dark.

A weak solution of the bark of the horse-chestnut.

A weak solution of sulphate of quinine, Le. a solution of the
common disulphate in very weak sulphuric acid. Various other
salts of quinine are nearly if not quite as good.

Fluor-spar (a certain green variety).

Red sea-weeds of various shades: a solution of the red colour¬
ing matter in cold water. If a solution be desired, a sea-weed
must be used which has never been dried. Sometimes even a
fresh sea-weed will not answer well.

A solution of the seeds of the Datura stramonium in not too
strong alcohol.

Various solutions obtained from archil and litmus (see Arts.
65 to 72).

A decoction of madder in a solution of alum.

Paper washed with a pretty strong solution of sulphate of
quinine, or with a solution of stramonium seeds, or with tincture
of turmeric. The sensibility of the last paper is increased by
washing it with a solution of tartaric acid. This paper ought to
be kept in the dark.

A solution, not too strong, of guaiacum in alcohol.

Safflower-red, scarlet cloth, substances dyed red with madder,
and various other dyed articles in common use.

Many of the solutions here mentioned are mixtures of various
compounds. Of course if the sensitive substance can be obtained
chemically pure it will be all the better.

Conclusion.

240. The following are the principal results arrived at, in the
course of the researches detailed in this paper :—

(1) In the phenomenon of true internal dispersion the
refrangibility of light is changed, incident light of definite
refrangibility giving rise to dispersed light of various refrangi-
bilities.

(2) The refrangibility of the incident light is a superior limit

ON THE CilANUE OF KKFKANUlttlMTY OF UHIIT.

399

to the refrangibility of the component parts of the dispensed
light,

(3) The colour of light in in general changed by internal
dispersion, the* new colour always corresponding to the new
refrangibility. It is a matter of perfect indifference whether the
incident rays belong to the visible or invisible part of the
spectrum.

(4) The nature and intensity of the light dispersed by a
solution f appear to la* strictly independent of the state of polariza¬
tion of the incident rays. Moreover, whether the incident rays be
polarized or unpolarized, the dispersed light offers no traces of
polarization. It seems to emanate equally in all directions, as if
the fluid were self-luminous.

(5) The phenomenon of a change of refrangibility proves to
be extremely common, especially in the case of organic substances
such as those ordinarily met with, in which it is almost always
manifested to a greater or less degree.

((>) It a (fords peculiar facilities for the study of the invisible
rays of the- spectrum more refrangible than the violet, and of tin*
absorbing action of media with respert to them.

(7) It- furnishes a new ehemienl test, of a, remarkably searching
character, which seems likely to prove of great value in the sepa-
rution of organic compounds. The test is specially remarkable for
this, that, it leads to the independent recognition of one or more
sensitive substances in a mixture of various compounds, and shows
to a great extent, before such substances have been isolated, in
what menstrua they are soluble, and with what agents tiny enter
into combination. Unfortunately, these observations for the most
part require sunlight.

(8) The phenomena of internal dispersion oppose fresh difli-
culties to the supposition of a difference of nature in luminous,
ehemieal, and phospliorogenie rays, but. are perfectly conformable

* [The statement was designedly limited to “a solution” (which I thought
might include glass, as a solution made at a high temperature) because it seemed
d priori very likely that there might ho a dependence in the case of a crystal; and
I think that I had already seen reason from observation to at least suspect that
such there was in the case of crystals of nitrate of uranium. Some time afterwards
almost marked dependence was observed in the case of platinocyanidos. ]

400 ON THE CHANGE OF REFRANGIBILITY OF LIGHT.

to the supposition that the production of light, of chemical
changes, and of phosphoric excitement, are merely different effects
of the same cause. The phosphorogenic rays of an electric spark,
which, as is already known, are intercepted by glass, appear to be
nothing more than invisible rays of excessively high refrangibility,
which there is no reason for supposing to be of a different nature
from rays of light.

NOTES ADDED DURING PRINTING.

Note A. Art. 2ffi

Shortly after the preceding paper was forwarded to the Royal
Society, I found M. Edmond Becquerel’s map of the fixed lines of
the chemical spectrum, which is published in the 40th volume of
the Bibliothique Universelle de Geneve (July and August, 1842).
I had seen in Moigno’s Repertoire d’Optique Moderne, that the
map had been presented to the French Academy, and naturally
felt anxious to obtain it; but not finding any further notice of it
either in that work or in the Coniptes Rend us, I supposed that it
had not yet been published. The principal lines in this map 1
recognized at a glance. M. Becquerel’s broad band I is my l ; his
group of four lines M with the preceding band forms my group m ;
his group of four lines Rf forms the first four of my group n ; his
line 0 is my n. It is only in the last group that there can be any
doubt as to the identification; but I feel almost certain that
M. Becquerel’s P is my o, and the next two lines, the last in his
map, are the two between o and p. It is difficult at first to
believe that the strong line p should have been left out, while the
two faint lines between o and p are represented, but the difficulty
is, I think, removed by considering the feeble photographic action
in that part of the spectrum. M. Recquerel expressly states that
lines were seen beyond the last he has represented, though they
were hardly distinct; and on comparing together his map, Mr
Kingsleys photographs, and my own map, I think hardly any
doubt can remain as to the identification.

ON TTTK CHANGE OF REFRANGIBILITY OF LIGHT. 401

I take this opportunity of referring to another very interesting
paper of M. Bccqucrel’s, entitled “Des effets produits sur les corps
par les rayons solaires,” which is published in the Annates de
Olrimie, tom. IX. (1843), p. 257, and with which I was not ac¬
quainted till lately, or I should have referred to it before. This
paper contains, among other things, an investigation of the effects
of transparent and coloured screens on the luminous, chemical, and
phosphorogcnic rays, in which it is shown that, notwithstanding
the great difference in the action of a given screen on the three
classes of rays when we study the effect of the incident rays as a
whole, its action is the very same when we confine our attention
to rays of any one refrangibility. Among the media employed by
M. Becqucrcl, arc some whose absorbing effect I have mentioned
in the present paper, as having been determined by methods
depending upon the change of refrangibility. In such cases my
own results, as might have been anticipated, are in perfect harmony
with those of M. Becqucrcl. With respect to the results at which
I have arrived regarding the nature of the phosphorogenic rays of
an electric spark, which are mentioned towards the end of the
paper, I have been in a good measure anticipated by M. Becquerel.
Yet T do not think that even he was aware that so much of the
effect of the spark was due to rays of such high refrangibility.

Note B. Art. 105.

I have since succeeded, by a particular arrangement, in seeing
so far into the “ lavender ” rays as to make out the groups of fixed
lines m, n,p by means of light received directly into the eye, and
even to perceive light beyond that *.

As to the colour of these rays when they are well isolated, I
think the corolla of the lavender gives as good an idea of it as
could be expected from the circumstances. They seem to me to
want the luminousness of the blue and the ruddiness of the violet.

* [The arrangement actually adopted was to form a pure spectrum with a
quartz train in the usual way, to isolate and at the same time condense a small
portion of the spectrum by a small quartz lens of short focus placed in or near the
pure spectrum, and to view the spot of light so formed from some distance behind
through a quartz prism applied to the eye. This prism was cut to show practically
single refraction, and its office was of course to remove to one side the scattered
light belonging to the ordinarily visible spectrum.]

S. III.

26

402

ON THE CHANGE OF REFRANGIBILITY OF LIGHT.

No doubt much error and uncertainty has hitherto existed both as
to the colour and as to the illuminating power of these rays,
because the gray prolongation of a spectrum formed on paper by
projection has been mistaken for the lavender rays.

Note 0. Art. 154.

On adding common phosphoric acid to a solution of nitrate of
uranium no effect seemed to be produced, but on examining the
vessel some days afterwards, a precipitate was found to have
fallen. This precipitate proved to be sensitive in a very high
degree.

Note D. Art. 158.

I have since observed in a mineral solution a system of absorp¬
tion bands so remarkable, and so closely resembling in many
respects those found in the salts of peroxide of uranium, though
they occur in a totally different part of the spectrum, that I think
no apology is needed for mentioning the circumstance. The
medium referred to is a solution of permanganate of potassa, in
fact, red solution of mineral chameleon. In order to see the
bands, it is essential to employ a dilute solution, or else to view
it in small thickness, since otherwise the whole of the region in
which the bands occur is absorbed. The bands are five in number,
and are equidistant, or at least very nearly so. The first is situated
at about three-fifths of a band-interval above D ; the last coincides
with F , or, if anything, falls a little short of it. The second and
third are the most intense of the set. I have carefully examined
the solution for change of refrangibility, and have not found the
least trace. Ferrate of potassa shows nothing remarkable.

By means of the bands just mentioned, the colour of perman¬
ganate of potassa may be instantly and infallibly distinguished
from that of certain other red solutions of manganese, the colour
of which some chemists have been disposed to attribute to per¬
manganic acid (see a paper by Mr Pearsall “On red Solutions
of Manganese,” Journal of the Royal Institution, New Series,
No. iv. p. 49).

ON THE CHANGE OF REFRANGIBILITY OF LIGHT.

403

Note E. Art. 171.

If we suppose the angle of incidence exactly equal to 45°,
assume $ for the refractive index of the fluid, and apply Fresnel’s
formulae to calculate the ratio of the intensity of light reflected at
the exterior surface of a bubble, and polarized in a plane perpen¬
dicular to the plane of incidence, to that of light similarly reflected
and polarized in that plane, we find 0‘228 to 1, a ratio which
certainly differs much from one of equality. But in order to
render the two intensities equal, it is sufficient to increase the
angle of incidence by only 3° 35'; and in fact, as a matter of
convenience, the position of the observer was usually such that
the deviation of the light was somewhat greater than 90°, and
therefore the angle of incidence somewhat greater than 45°.

Note F. Art. 191.

I have since received a slab of glass of the kind here
recommended, which has been executed for me by Mr Darker
Df Lambeth, and which answers its purpose admirably, the
medium being eminently sensitive. Besides its general use as
x screen, this slab, from its size and form, has enabled me to
brace further than I had hitherto done (Arts. 75, 76) the
connexion between certain fluctuations of transparency which
die medium exhibits and corresponding fluctuations of sensi¬
bility.

Note G. Art. 192.

Paper washed with a mere infusion of the bark of the horse-
chestnut is quickly discoloured; but a piece washed with a
solution which had been purified by chemical means remained
white, and proved exceedingly sensitive.

Note H. Art. 204.

I have since ordered a complete train of quartz, of which a
considerable portion, comprising among other things two very fine
prisms, has been already executed for me by Mr Darker. With
Lhese I have seen the fixed lines to a distance beyond H more

26—2

404

ON THE CHANGE OF REFRANGIBILITY OF RIGHT.

than double that of p ; so that the length of the spectrum, reckoned
from H , was more than double the length of the part previously
known from photographic impressions. The light was reflected
by the metallic speculum of a Silbermann’s heliostat, which I
have received from M. Duboscq-Soleil. With the glass train the
group p was faint, but with the quartz train there was abundance
of light to see not only the group p, but the fixed lines as far as
Hpl, or thereabouts. From the group n to about the middle of
the new region, the lines are less bold and striking than in the
region of the groups II, l , m, n, but the latter part of the new
region contains many lines remarkable both for their strength and
for their arrangement. I hope to make a careful drawing of these
lines as shown by the complete train with a summer’s sun.

I have some reasons for believing that the photographic action
of these highly refrangible rays is feeble, perhaps almost absolutely
null. In the second of the papers referred to in Note A (p. ;}(>()),
M. Becquerel describes an experiment in which a prism of quartz
was employed to form a spectrum ; and yet the impressed spectrum
formed by rays which had traversed the quartz alone was hardly
longer than that formed by rays which, in addition to the quartz,
had traversed a screen of pure flint-glass a centimetre in thickness.
It is possible, I am inclined to think probable, that glass made
with perfectly pure materials would be transparent like quartz,
but all the specimens I have examined were deeidedly defertive
in transparency. Besides, M. Becquerel, who may bo allowed to
be the best judge of his own experiments, considered the result-
just mentioned as a proof that the impressed speetrum formed by
rays which had traversed quartz only did not extend, except a
very trifling distance, beyond that formed by his train of glass ;
and yet his map, formed by means of the latter, does not take in
the line p.

However, among the multitude of preparations capable of
being acted on by light, it is probable that there may lx* some*
which are acted on mainly by rays of unusually high refrangibilif y,
and which, on that very account, would not be suitable for the
ordinary purposes of photography. With these it is possible
that the new region of the solar spectrum might he taken
photographically.

ON THE CHANGE OF 11EFRANGIBILITY OF LIGHT.

405

Note I. Art. 213.

I have wince examined the salt, or product, whatever it may
be, in the dry state, and under more favourable circumstances, and
have found it sensitive, though not by any means in a high degree.
It exhibits also the absorption bands which seem to run through
the salts of peroxide of uranium.

hi connexion with the insensibility of a solution of nitrate of
uranium in ether, it seems interesting to mention a fact which I
have since observed, namely, that the sensibility of a solution of
nitrate of uranium in water is destroyed by the addition of a little
alcohol.

Note J. Art. 217.

On repeating this experiment on a subsequent occasion, I
could not satisfactorily make out the difference of character of a
strong and of a weak spark from the prime conductor, perhaps
because the machine was in less vigorous action; but the difference
between the effects of a mere spark and of the discharge from a
Leyden jar was plainly evident. I would here warn the reader,
that in order to perform the experiment in such a manner as
to obtain a striking and perfectly decisive result, it is essential
to employ an excessively weak solution. The reason of this is
(widen t.

A severe thunder-storm which visited Cambridge on the
evening of July 16, 1852, afforded me a good opportunity of
observing the effect of lightning on a solution of quinine, and
other sensitive media. From the copiousness of the dispersed
light, it was evident that the proportion of the active, and
therefore highly refrangible, rays to the visible rays was very
far greater in the radiation from lightning than in daylight.
A difference of character was observed between the effects of
a weak distant flash, and of a bright flash nearly overhead,
similar to that which has been described with reference to the effects
of a spark from a machine, and of the discharge from a Leyden jar.
In artificial discharges, the stronger the spark the more the rays of
excessively high refrangibility seem to abound, in proportion to
the whole radiation. Now a flash of lightning is a discharge

406

ON THE CHANGE OF HEFIUNGIBILITY OF LIGHT.

incomparably stronger than that of a Leyden jar. It might
have been expected, therefore, that the radiation from lightning
would be found to abound in invisible rays of excessively high
refrangibility. Yet I could not make out in a satisfactory manner
the absorption of the rays by glass, even by common window-
glass. I do not wish to speak positively regarding the result
of this observation, for of course observations with lightning
are more difficult than those made with a machine which is
under the control of the observer. Yet it did seem as if the
spark from a Leyden jar was richer than lightning in rays of
so high a refrangibility as to be stopped by glass. If this be
really true, it must be attributed to one of two things, either
the non-production of the rays in the first instance, in the
case of lightning, or their absorption by the air or clouds in
their passage from the place of the discharge. If they were
not produced, that may be attributed to the rarity of the air
at the height of the discharge, that is, at the height of the
thundcr-cloud. No doubt the metallic points of the discharger
belonging to the electrical apparatus may have had an influence
on the nature of the spark; but I am inclined to think that this
influence, so far as it went, would have acted in the wrong
direction, that is, would have tended to produce rays of lower, at
the expense of those of higher refrangibility.

Note K. Art, 220.

My attention has recently been called to a paper by M. llriirke
(Poggcndorffs Annalen, R V. (LS45) S. add), in which hr describes
some experiments which show that the different parts of tin* eye,
and especially the crystalline lens, arc far from transparent with
respect to the rays of high refrangibility. The eyes employed
were those of oxen and some other animals; and the inquiry was
carried on by means of the effect which light that had passed
through the part of the eye to be examined produced on a film of
tincture of guaiacum that had been dried in the dark. ()f course
the phenomena described in the present paper afford peculiar
facilities for such an inquiry, and I had frequently thought of
entering upon it, but have not yet made any observations.
Independently of the facility of the observations, and the ad-

ON THE CHANGE OF REFRANGIBILITY OF LIGHT. 407

vantage of being able to examine readily light of each degree
of refrangibility in particular, the results obtained by means of
sensitive media seem to be more trustworthy on this account,
that it would be possible to employ fresh eyes. The experiments
of M. Brticke necessarily occupied a considerable time, and it may
be doubted whether the eye, especially after dissection, might not
have changed in the interval, and whether the results so obtained
are applicable to the eye as it exists in the living animal.

INDEX TO THE PRECEDING PAPER.

N.B. The futures refer to the articles, the letters to the original notes,
2>p. 400—407.

Absorbing and reflecting power, deter¬
mination of, with respect to invisible
rays, 194—196, 201—204,

Absorption, on the cause of, 237, 238;
connexion of, with internal dispersion,
59, 60, 63, 71, 76, 120, 126, 140, 148.

Air expelled from water, 172.

Appearance of highly sensitive media,
27, 29, 164.

Archil, 65—71.

Canary glass, 73—77,I<\

Chemical applications of internal dis¬
persion, 67—70, 205—214.

Clearness of sensitive fluids accounted
for, 86.

Coloured glasses, fundamental exjM.Ti-
mont with, 7; nature of the blue re¬
flexion in orange glasses, 79; examined
for dispersion, 167, 168.

Colourless glasses, internal dispersion
in, 78; defect of transparency of, 22,
202, 217—219, A.

Colours of natural bodies, 174 17H.

Concentration and dilution, effect of,
185—188.

Crystals, natural, 165, 166 (see Fluor¬
spar, Uranium).

Datura stramonium, solution, 43; paper,
95; capsules, 117.

Dispersion, true and false, defined, 25;
distinguished, 26, 29; cautions, 169- -
173; usual features of true, 44—46;
on the cause of true, 226—236; nature
of false, 179; instances, 181; appli¬
cations, 180, 182.

Epipolic dispersion, 1; explanation of, 6.

Esculine, 31, 212.

Explanation of terms, 21—30, 101.

Eye, opacity of the, 220, K.

Fixed lines of the invisible, rays, ex¬
hibited, 16; description of the, 21 23,

A, II; effect of viewing through a
prism the fixed lines shown by means
of sensitive media, 17, 89 93, 100.

Flames, effect of various, 197 - -209.

Fluor-spar, 32 36.

Groundsel, petals of the purple, 120.

Guaiacum, solution, 37 41; washed

paper, 98; solution of, used hh a test
object, 200.

Heat, effect of, on the sensibility of
glass, Ac., I Hi.

Horse-chestnut, 31, 212.

Illuminating power of the highly re¬
frangible rays, U) 1, 105,

Internal dispersion, 2.

Lavender rays, 105, L.

Leaf-green, absorption, 17 52; internal

dispersion, 53 61, 97, 118, 121.

Lightning, effect of, J.

Linear spectrum, 107 109; n suits ob¬
tained with a, 113 136, Ac.

List of highly sensitive substance.., 239.

Litmus, 72.

Mereurialis pemmis, 62 61.

Methods of observation, gcneial, 13, 17,

110 , 112 .

Mutual action of incident lays, negative
results with reference to the, 215, 216,
235, 236.

Permanganate of potassa, ah. orption, I>.

Phosphorescence compared with internal
dispersion, 220 225.

Phosphorogenic rays of an electric , pat k,
nature of the, 219 221, A.

Polarization, absence of, in di: perse*l
light, 1, 15; of the incident, iays a
matter of indifference, 20,

ON THE CHANGE OF REF11ANGIBILITY OF LIGHT. 409

Polarized light, direction of the vibra¬
tions in, 183.

Quartz, transparency of, 202, 217—219;
train, 204, II.

Quinine, absorption of the violet by a
solution of sulphate of, 11; internal
dispersion in a solution of, 14—20;
strong affinity of, for hydrochloric,
hydrobromie and hydriodic acids,
205—211.

Itays, course of, exhibited, 193.

lleflecting power (see Absorbing power).

Kefrangibility of dispersed light, lower
than that of the incident, 80, 102,
230 —230; nature of the, 81, 82; il¬
lustrated by a Hiirfaco, 84, 85.

Results, principal, 240.

Screen, on the choice of a, 189—192, F,

G.

Sea-weeds, red, 121—126.

Strata of equal dispersion in crystals,
167.

Test objects, 110, 114, 200.

Triangle, experiment with a paper, 58.
Turmeric, tincture of, 42; paper, 87—91.
Uranium, salts of the peroxide, &c.,
157—162, 213, C, I (see also Canary
glass); delicate test of, 159; absorption
of light by salts of protoxide of, 160,
163.

Washed papers, 87—98.

ADDITION TO THE RRECKIM N(! RARER.

[For a long time, in fact from before 1867, I have taken a
different view of the nature of tin* phenomenon from that indieatcd
in Arts. 63 and a few following. I was h*d to it by a phenomenon
I casually noticed in making observations on the thioresrenee of
various glasses. The sun’s light was rebooted horizontally into a
darkened room, passed through a convex lens, and with or without
previous filtration by passing through a blue glass plaeed at or
near the focus, was admitted info tin* glass to he examined. In
general the fluorescence when examined in this way showed no
appreciable duration, hut in one class of glasses when tin* glass
was merely moved sideways by hand somewhat rapidly a<To» the
incident pencil, a luminous trail was seen extending bom the
focus at the moment into tin* part of the glass on which the focus
had previously fallen. Tint colour of the fluorescent rone as a
whole wdien the glass wars at rest, and that of tin* root of the trail
(or part nearest to the focus) when the glass w\*n in nnam, varied
with the composition of fin* glass, but the dmt ribution of {he,
fluorescence in tin? spectrum wdien the glass wars oxamined in an
intense and fairly pun* spectrum was the same or \cry nearly so,
and was that characteristic distribution alluded fn in ^ 78. 1 may

mention that I have eireumstantial evidence that the » ubstanee
in the glass to wdiich tliis eife«*t was due w as maugauc e in a lower
state of oxidation than that, wdiich gi\ cs (In* purple c»»lnur. Now
the colour of the trail changed from its root, to its end, 1 >eetaniug
reddish at last, in such a manner as to indicate a den easing mean
iehangrbility, excc.pt m one cast*, t hat of a pinephat ie , j > la.- > s
prcpaied by the Rev. \V. V(*nn»n-1 huvourt, where the colour was
led to start with, so that there was no opportunity to change.

Ihc change of colour with the lapse of time since excitement
of the glass led me to regard the alteration in the molecular
distm banco as brought about in the following manner. The

OX THK CHANGE OF 11EFRANGIBILITY OF LIGHT.

411

incident vibrations of the ether agitate the sensitive molecule or
complex molecule, and this agitation gradually spreads outwards
by the molecular forces into the complex system formed of the
sunounding molecules and molecular groups, which may be mostly
themselves insensible, very much in the manner in which if a
vcr y mi *nute portion ot a metallic mass be supposed strongly
heated to start with, the heat is rapidly diffused into the neigh¬
bourhood by conduction ; and if the heated spot is at the surface,
so as to permit of radiation outwards, the radiant heat is of a
mean ref Eligibility which decreases as the time goes on. Indeed
the phenomenon of fluorescence, which is a brief phosphorescence,
seems to be closely akin to the very familiar one of the heating
of a body by sunshine, and the consequent emission of heat-rays
of low refrangibility.

Being desirous of seeking an analogy in some simple dynamical
problem which could be actually worked out, I took the case of
an infinite weightless flexible and inextensible stretched string,
loaded at equal intervals with equal masses, and supposed one of
the masses permanently acted on by a small transverse disturbing
force expressed by c shinty t being the time, and I demanded the
permanent state of motion of the masses. It proved to be of a
different character according as the frequency of the disturbing
force was greater or less than that corresponding to the shortest
of the periods, infinite in number, belonging to the natural oscilla¬
tions of the loaded string when no disturbing force is acting.
Take the mean position of the mass on which the disturbing force
acts for the origin of abscissae. The motion will of course be
symmetrical on the two sides of the origin, and it will suffice to
refer to that on the positive side. When the period of the force
is greater than the critical period, the disturbance extends to
infinity. It is of the nature of an undulation propagated outwards
from the origin, and the various masses will at any moment lie on
a curve of sines. But when the period of the force is less than
the critical period, the disturbance instead of extending to infinity
decreases on receding from the origin, and the masses at any
moment lie in a curve derived from a fixed curve of sines by
diminishing at a given moment the ordinates in geometric pro¬
gression as we recede from the origin, while as the time changes
all the ordinates vary as sin nt

412

ON THE CHANGE OF REFKANGIBlLITY OF LIGHT.

Suppose now that the system is at rest and that the disturbing
force begins to act. A motion will be produced gradually tending
to become that permanent kind of motion above referred to. Jf
the period of the force be greater than the critical period, no
significant local disturbance can result, because the disturbance
near the origin which the force tends to produce is carried away
in both directions. But when the period of the force is less than
the critical period the local disturbance mounts up continually,
tending towards its permanent state.

Suppose now the disturbing force, after having acted long
enough to bring the motion into its permanent state, were then
to cease. In the first case there would indeed be a change of
motion in the neighbourhood of the origin; but as the excursions
of the masses would be but small, for the reason already mentioned,
the effect of the change would be quite insignificant. But in the
second case the disappearance of the force would leave the masses
in the neighbourhood of the origin in a condition of displacement
or motion which is not, as in the former case, insignificant, and
which may be thought of as an initial disturbance in a system
now left to itself. This disturbance would gradually widen out so
as to involve a continually increasing group of masses, and the
variation of the motion as at a given moment we proceed along
the string would become less and less sharp as the time progresses,
so that the disturbance if harmonically resolved would give a
result in which the elements of the resolution corresponding to
longer periods would acquire an increasing relative importance.

The general analogy of these dynamical results to tho plmim-
mena of fluorescence will I think be readily perceived. Tim
periodic force acting on one of the masses of our loaded string is
analogous to the action of the incident ethereal vibrations on a
constituent part of a complex molecule or molecular group. Tim
considerable but finite regularity of the disturbing foreu is
analogous to what we must suppose to be going on as regards tint
ethereal vibrations in a small portion of a nearly pure spectrum.
The agitation of the masses when the disturbing force; erases and
the loaded string is left to itself is analogous to the agitation of
the excited molecules when one series of regular incident, ethereal
vibrations comes to an end and is replaced by aimther such scrie*s,

ON THE CHANGE OF REFRANGIBILITV OF LIGHT. 413

or a secular change takes place in the incident vibrations which
is equivalent to a succession of such independent series.

We are far from being able to subject molecular disturbances
to strict mathematical calculation; for in the first place we are
ignorant of the molecular structure to which the calculations
would have to be applied, and in the second place even if we did
know the structure, the calculations would very probably prove
to be too complicated to be satisfactorily carried out. Neverthe¬
less the calculation in some assumed system of a tolerably simple
character which possesses some features in common with that
which is supposed to lie at the base of some observed phenomenon
may not be without use, as indicating in what direction we are to
look, or are not to look, for an explanation of the phenomenon.
In the present case if we vary the problem by merely supposing
that the equidistant masses are of two magnitudes, coming
alternately, a result is obtained having some features strikingly
resembling those of absorption.]

INDEX TO VOLUME III.

[The references are to pages.]

aalyser, a new elliptic, 197.

rc of vibration of pendulums, effect of

viscosity on, 111.

lily’s pendulum experiments, discus¬
sion of, 70.

sssel’s pendulum experiments, discus¬
sion of, 92.

aange of rcfrangibility of light (ab¬
stract), 259 ; (paper in full), 267. For
index see p. 408.

.ouds, explanation of apparent suspen¬
sion of the, 10, 59.

floured rings, experiment showing
fixed in air, 159, 167, 180.
flours of natural bodies, 350.
flours of thick plates, 155.
imposition and resolution of streams
of polarized light from different
sources, 233.

iulomb’s experiments on viscosity, 9,
21, 97.

rystals, conduction of heat in, 203;
fluorescenco in, 242, 351, 399.
finder in a viscous fluid, oscillating,
38; moving uniformly, 02.
issipation of energy due to viscosity,
07; dissipation function, 09.
ubuat’s experiments on resistance, 107.
lliptically polarized light, experimental
determination of elements of, 197.
Lliptically polarized light, distinction
between, and common light, 235.
pipolic dispersion of light, 259, 207.
iames, feebly illuminating, richness
of, in rays of high rcfrangibility, 373.
luorcscence, 289 ; cause of, 271, 388,

410; comparison between, and phos¬
phorescence, 385.

Heat, conduction of, in crystals, 203.

Instability of motion of a viscous fluid
past an infinite cylinder, 62.

Intensity, total, of interfering light, 228.

Interference, absence of, in two scattered
streams of light from the same source,
187.

Internal friction of fluids (see Viscosity).

Opposite polarization, definition of, 239.

Pencil of rays, delicate mode of exhi¬
biting a, 370.

Pendulums, effect of internal friction of
fluids on motion of, 1.

Polarized light, direction of vibrations
in, 361.

Radiation, effect of, in stifling sound,
142; on the velocity of sound in a
tube, 149.

Sound, possible effect on, of radiation
of heat, 142.

Sphere in a viscous fluid, effect of vis¬
cosity on an oscillating, 22; on a
uniformly moving, 55.

Stifling, theoretical, of sound by radia¬
tion, 142.

Tomlinson’s experiments on viscosity,
139.

Uranium compounds, relation between
dark bands of absorption in, and
bright bands of light emitted by, 345,
346.

Viscosity of fluids, effects of, on the
motion of immersed bodies, 1.

CAMBRIDGE: PRINTED BY J. AND C. F. CLAY, AT THE UNIVERSITY PRESS.