FIGURE AND STABILITY OF A LIQUID SATELLITE. 
189 
we 
find 
(eE) = (f^a 3 ) 2 
1 + 
+ 
+ 
+ 
+ 
2.5 r 3 
3 
2 3 .5.7r 5 
3 
V P.+ l + ^f ^5+1 
K 
*.(3 + 2 + 3^ + ^8_ + ^ + 3 
K K 
2 2 .5 2 r 5 
1 
2 4 .3.7r 7 
9 
2 4 .5.7r 7 
M^ + ? + >? +3 
^ 6 (^+l + | 2 + 5 ) + ^f|e + |i + -+ 5 
K K 
K 4 K 
k 2 Kl 
5 I 2 2 1 
"YUi + F* + + Fa + ^ + 5 
kK K kK K k 
- 4 K 2 ' K 4 ' k 2 K 2 
(14). 
The first term in this expression is that which was called above (eE) ]} and the rest 
constitutes (eE) 2 . 
If the body E were a sphere, the only portions of (14) which would remain would 
be the parts of the expression independent of 
With the object of effecting certain differentiations hereafter, it is desirable that 
the formula for (eE) should be expressed in terms of the semi-axes a, b, c and A , B, C. 
In accordance with the notation used elsewhere, we have 
a = 
k cos 
7 
sin j3 
b = 
h cos /3 
sin /3 
c = 
sin (3 
, where sin /3 — k sin y, 
A — , B = ^-. cos B , C = , where sin B = K sin T. # 
sm sin d sin U 
The result of the translation into this other notation is as follows :— 
(eE) = (|-7rpa 3 ) 2 ^ + { 1 + 7rT~5 [2c 2 -« 2 -// + same in A, B, C) 
+ >V1 3 ■■ . [3 (a 4 + b 4 ) + 8c 4 — 8c 2 (cr + b 2 ) + 2 a 2 b 2 + same in A , B, C] 
2 .5. i v 
+ -^- 5 [2 (AW + B 2 b 2 + C 2 c 2 ) + (A 2 +B 2 +C 2 ) (a 2 + b 2 +c 2 ) 
-5C 2 (a 2 + b 2 ) -5c 2 (A 2 +B 2 ) + 5 C 2 c 2 ] 
+ 1 [ 16c 6 - 5 (a 6 + E) - 24c 4 (a 2 + b 2 )+ 18c 2 (a 4 + b 4 ) 
— 3 a 2 b 2 (a 2 + !/)+ 12 a 2 b 2 c 2 + same in A, B, C ] 
+ -jL—[-A 4 (5a 2 +b 2 -6c 2 )-B 4 (a 2 +5b 2 -6c 2 ) 
2 4 5 L ' ' 
- 8 C 4 (a 2 + b 2 - 2c 2 ) + 4 B 2 C 2 (a 2 + 3& 2 — 4c 2 ) 
+ 4 C 2 A 2 (3a 2 + b 2 —4c 2 ) - 24 IB 2 (a 2 + 6 2 + 2c 2 ) 
+ same with small and large letters inter¬ 
changed] l .. . . (15). 
* The fact that capital /3 is nearly the same as B must be pardoned; it cannot, I think, cause any 
confusion. 
