ON THE MOTION OF PENDULUMS.                          35
For a given fluid and a given time of oscillation, both Jc 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,
a                  \      ma           \      ma
-l+=a?c (55), a                   \          maj         \          maj
(-56), =0  (57).
Putting a?cK for af^(a) + 2/_2(&), which is the quantity that we want to find, we get from (38) and (54)
(58).
^    ^J
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, and D changed. Hence — A, — 0, — D are the same functions of b and a that A, C, D are of a and b. It will also assist in the further elimination to observe that 0 and D are interchanged when the sign of m is changed. The result of the elimination is
K=l-^~         qM)-qfta)
2mV' IZmab + £ (a, J) -f £ (6, a)
the functions £ 77 being defined by the equations ^ (_a> If) = (_mV 4. 3_ma + 3) (_m^2&^2 - 3mb + 3) e^m(?J"^a)
, f (a, b) ={J(m^26^2~3m6+3)--a(mV+3ma+3)}6^m(z'-"^)J......^(    ^j'
3—2