OF EQUILIBFJUiM OF A KOTATING MASS OF LIQUID. 313 
All these integrals are exj^ressible in terms of the elliptic integrals 
"=ri4- 
It will, however, be found that in fact the coefficient of IT vanishes in every case. 
The cases of f = 0 and f = 1 are very simple, and we have 
13,Q, = Av.Ar - 
siir y \ K 
= X cot" y (\ A tan y — , 
' 1C 
Pi'Q,' = A-(- f F + ~E- 
sin- y \ K- a:-/c - K j 
It is possible by direct differentiation to verify the following results, although the 
verification will be found pretty tedious. 
rsin^Ap 
I ATa = 
J A/A 
(2 - .-.ghr/ - - 2r/) 2q'-^ - 1 1 Asmjrcosf 
- r/) 2 qY~ 2 q^q'^(K^-cf) ^ 9 
2</\.^ - 2QAf ’ 
f 
sin^ ^fr T . 1^.1 — 2k^ . 1 
c/Q = — - F + 
w-) /.I 
K-’fC 
E + -A tan xfj, 
cos® -yjrA 
fsin^^fr J 2 „ 1 + /c'® „ sin-v/r cos a/t 
1 ^ #= --A’d- E- -, 
f siu^ Ip 7; _ 1 77 t U ^ I tail Ip I • 0 ,7 
)cos=fAU'/'- *v. [- - (I +'<-)sii>->H- 
These are all the integrals needed for the harmonics of tlie second degree. In the 
case of the first we have 
= <r 
O ‘■>x^2 
Z — 
1 - 2^® ■ 
Thus the coefficient of IT vanishes and the results are 
/cAj^ 
aiid ^2^^) = 7^ 
1 - 2 q- _ 1 - 2i/ (1 - 2 q~)A sin 7 cos 7 
siu*7L 2qY~ ^ "T 2i/®(/OAp 
(-) {-) = 
K COS"' 7 
1 ^ ,1 - 2/^2 „ , 1 
— A H-AA.-' + Ti A tan y 
K 
0 /o 
K-'K " 
siii*^ y [_ fc 
a:A® r 2 . 1 + /c'® sill 7 cos 7* 
} 3 o> (,-) ©»' (v) = - - 7 F + F 
T- - w ~ \ / Slid 7 /c‘ ® 
a:*a: 
K cos® 7A® 
.-.A2 r 1+y_ ^ 
0 /O .0 /.t ^ I 
si id 7 
/c®/c'® A 
tan y {2 — {1 + K-) sin® 7)' 
K~K ■ 
k'-^A 
In the first of these cf = -i [l + k® d- (1 — k'k'^)"] and k~ = 
2 - 39 ® 
2f 
VOL. cxi’vnr.— a. 
