FIGURE AND STABILITY OF A LIQUID SATELLITE. 
93 
(eE) 2 = 
/ 4._„ q 3\2 k 
("3 7 T P A ) 
(i+x)H2^3[ F f? +1 ) + 7e (i? +1 
+ 
+ 
+ 
+ 
2 3 .5 .lr h 
3 
* 4 (| + f + 3 Mf; + l +8 
+ 7T5 9-i + 3 
2 2 .5 V 5 Mk 2 K 2 K- k 
i# (? + ? + ? + 5 ) + ^ 6 (l i+ l i+ l +5 
2.3.7r 
2 4 .5.7F 
7 2 j^4 ( 5 . 1 , 2 2 1 
4 ^-(vK‘ + K‘ + ?i? + K 2 + ? +5 
+ ^<A + ? + il? + l + ? +5 
( 21 ). 
10. Determination of the Forms of the Ellipsoids. 
We have obtained in the last section the expression for V, the lost energy of the 
system. 
The harmonic deformations of the ellipsoids being of orders higher than the second 
do not enter into the moment of inertia to the order of approximation adopted. 
Hence the moment of inertia about the axis of rotation, which passes through the 
centre of inertia of the system, and is parallel to the a and A axes of the ellipsoids, is 
given by 
I — i}7rpa' s 
1 + X 
Ar 2 
(i+WJ 
• ( 22 ). 
If f denotes any one of the parameters by which the system is defined, the 
condition that the figures shall be in equilibrium is 
0F 
3/ 
07 
3/ 
+i ( ° — — d. 
The parameters defining the system may be taken as r, the distance between the 
two centres, cos y, cos ,8 for the smaller ellipsoid and cos F, cos B for the larger one. 
Besides these we have the coefficients f \ F* of the harmonic inequalities of order i 
and rank s on the two ellipsoids. 
For convenience write 
a = cos y, i) = cos ft. 
These letters are chosen on account of the association of cos y, cos ft with the 
semi-axes a, b of the smaller ellipsoid e. It is unnecessary to adopt a corresponding 
notation for the ellipsoid E, because, when the problem is solved as regards e, it 
affords the solution for E by symmetry. 
Since 
F cos ft cos y cosec 3 ft — \a 3 /(l +X), 
2 c 
VOL. CCVI.-A. 
