EQUILIBRIUM OF ROTATING MASSES OF FLUID. 
395 
In order to find the disturbance of the spheroid a, we add the perturbing potential 
to §7 ra 3 /r, give r its value (53) in this term, put r = a in the perturbing potential, 
and make the whole potential constant by equating to zero the coefficients of each 
harmonic term. 
We will begin by putting r/a = 1 — ^-e 8 hvjr*, and considering only the perturbing 
potentials (54) and (58). We have then, for the coefficient of 8 2 w 4 /r 2 , 
2 7ra . qCO CL “ | q(0 OL • 
Now, with the value of e in (37), 
Itto 3 . and - 2 \ — | ^ = 0 . 
Hence the coefficient of Shvjr 2 vanishes, and the term e in (53) has been properly 
chosen to satisfy the perturbing potentials (54) and (58). Following the similar 
process with the remaining terms of (53), and equating to zero the coefficient of 
8 hvjt+%, we have, from (55'), (56), (57'), 
or 
2k + 1 
2k - 2 
n \ 3 i = =o 
k + 7! 
v-i 
cl i = 2 i — 2! k + 2 ! i — 1 
M; = 0 , 
2 - 
h + 7 ! 
p-i 
= 2 i - 2 ! 7; + 2 ! 7-1 
Mi. . 
(59) 
By symmetry the condition that the spheroid A may be a level surface is 
7 
,£ — 1 
mi. 
(60) 
k + r ! 
v-i 
Multiply both sides of (60) by f (jy “— anc ^ perform X on the 
whole, and substitute from (59), and we have 
m , _ j_ £ /Af = _i. e WiW'j * + jw 
10 \c ) 10 \c/ 8 \c / = 2 r — 2 ! 7; -h 2 ! r — 1 
r + 7! h + r ! 
I /3\3 ( A 'f T T ilk + r! _ r '' _1 fihl 1 
^ ^ V/ w r = 2 i = 2 i - 2 ! r + 2 ! r - 2 ! h + 2 ! r - 1 7-1 
mi. 
Now let us write 
{k, r}= 2 l + r - 
r = 2 T 
- 1 \Tc + 2 ! 
F 
>— 1 
r + 7! k + rl 
- - r t-llr + 2!r-l!4+ 2 
, r '-‘ J 
so that (61) may be written 
(61) 
(62) 
r/ u — io e ( c j + To' e 
M\ 3 . i 3 /<U 3 
r) + m*)'(7ty IVALI (<53) 
