LIQUID SPHEROID OF FINITE ELLIPTIOITY. 
199 
In the waves produced the values of n, s, k will be the same. 
Substituting from (21), (52), (53), (55) in (51), we obtain 
+ 47rpy cot a cosec^ a (^,j) q^Q — C(Co) (Co)} 
= . . (56). 
This equation, combined with (39), suffices to determine the unknown constants 
A,/, C,/ in terms of the known coefficient and thus the amplitude of the forced 
oscillation is determined in terms of that of the disturbing force. 
10 . The most interesting point is to determine C/, in order to find the height of 
the corrugations on the surface. This plan has, moreover, the advantage that, in 
considering the effect of several disturbing forces of different periods, we may add 
together the elevations [h) due to the separate forces, whereas, in determining the 
value of ifj, the terms having different periods are referred to different auxiliary 
systems. Substituting for A/ in terms of C./ from (39) in (56), and writing, for 
brevity, 
K/ (Co) = Pi (Co) qi (Co) - 4" (Co) (Co). (57), 
M = cot a cosec® a = mass of spheroid . . . (58), 
we find, after several reductions, the required equation for C/, viz., 
( Co) (^o) 
s D*P„ (vo)l{/c — 1) + sec^« . j'oD^ + ffy, 
= W; 
(n, k) 
(59), 
in which it must be remembered that 
K cos a 
~ V" (1 - sill® a) 
(15). 
The Period-Equations for Free Waves. 
11 . If the oscillations of the liquid be free, we must put W‘(„_ equal to zero in (59), 
and we therefore obtain 
K/ (cot a) 
4 :Kq^ (cot «) D*P„ (i/q) 
sD^P,j(i'g)/(/c —1) + sec^ a . D^'+iIb (i^o)/« 
(60), 
which, together with (15), determines the admissible values of k and Vq. In reducing 
(60) to a rational algebraic equation for k we must distinguish three cases. 
I. Let s = 0, and let n be even. Then we know that P„ (^^q) and DP„ (vq) contain 
only even and odd powers of Vq respectively, and, therefore, that DP„ {vq) is divisible 
by Vq. Multiplying (60) by J)Vn(v,^lvQ, we find (writing K„ for K„°) 
K„ (cot a) DP„ {vq)Ivq — Aq^ (cot a) (l — k® sin® a) P„ (vq) =0 . . (61). 
