248     ON  THE  COMPOSITION  AND  RESOLUTION  OF  STREAMS  OF
spar or other crystal, cut for shewing rings, followed by a Nicol's i                        prism. The plane-polarized pencils respectively stopped by and
^1                        transmitted through the Nicol's prism consisted, on entering the
crystal, of pencils elliptically polarized in opposite ways; and i                        the nature of this elliptic polarization changes in every possible
i '                      manner from one point to another of the field of view. If these
i                      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 Nicol's prism or other analyzer.
11.   theorem.    Let a polarized stream be resolved into two 1   j                           oppositely polarized  streams; let the phase of vibration of one
I  m ;                          of the streams be altered by a given quantity relatively to that
I1  j i                          of the other, and let the streams be then compounded.    If the I ^: I^f                         polarization of the original stream be now changed to its oppo-I   i)                         site, the polarization of the final stream will also be changed to 1   I! ,                       its opposite.
*f   11 i                             The straightforward mode of demonstrating this theorem, by
!f   !                           making use of the general expressions, would lead to laborious
f   \                           analytical   processes,  which   are  wholly   unnecessary.     For   the
^l|   *'                         formulae which determine the components of a given stream are
|    I                         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 formulas themselves, or the geometrical
j * _(!                         conditions of which the formulse are merely the expressions, are
. _( ,                         satisfied.
_1 i'                                 For shortness' 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 are concerned to be
turned in azimuth through 90°. This will not affect the geometrical relations connecting components and resultants. Let X', 0', E', o'', e, Y be the streams which X, 0, E, o, e, Y thus become. The streams 0'', 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 0' to c/ the ;                       phase is retarded by p. Now conceive the direction of motion of