FIC4URE AND STABILITY OF A LIQUID SATELLITE. 
199 
Therefore 
a sW = -M’(iw- 
On adding together (28) and (29), we find 
3a S = A- 3 (6 3 —2a 3 —2c 2 +rc 2 )] , 
3l>^= ( ^ a T fr ~ ? [-!(3a, , -i4)+ A» (a >_2&»_2c»W)] . 
. . (29). 
By means of (i) and (i)' we find the differentials of the moment of inertia I ; 
they are 
rl J \ 
Ub 2 +c 2 ), 
Sr — = - ^Trpa? 
da 3 r l+\ 
,3 ^ 
3i) f = (W-O. 
Then, since |cu 2 = ^irpsd. ~ (1 + £), 
!, a = '^ ,rpa ^(TTx) i ' 5? (1+x)(1 (6!+<;2) > 
*S = ^) a ( I ^ ? -^0 + X)(l + 0(2^). 
3.2, 
o CO 
3 _2 
Now the equations for equilibrium for the parameters a and ft are 
c/ b -i 2 dl A d V i 2 dl A 
* + ^35 = 0 ’ - + i“ 3 ^ = 0 ' 
Therefore 
k 
db 
db 
-3\(3A 1 1 -m)+ 4 ( [6 2 -2a 2 -2c 2 +rc 3 - (1 + X) (1 + 0 (6 2 + c 2 )] = 0 
3X(3a , -14)+ 4[a J -2^+2c ! + a -c s +(l + X)(l + 0(25 J -c i, )] = 0 
(30). 
Subtracting the second of these from the first and dividing by 9\, we have 
■Hi 1 -A. 1 = g^+,[« 2 -i» 3 + ^ (1 +X) + . . . . (31). 
