FIGURE AND STABILITY OF A LIQUID SATELLITE. 215 
With the lower sign for s = 2, 
q' 2 (K 2 -q 2 ) =_1^(1 + 0/c' 2 -*k' 4 ...).(55). 
The three last expressions (53), (54), (55), give the two values of ^/(l) C 3 s (^ 7 t). 
We now turn to the functions ® 3 S . The general definition of § 5 was 
®/ = j LWW ^‘WJpda, 
where M was the mass of the ellipsoid. Hence in the cases under consideration 
*•* = 4 l I*-’W ^ ^ ( S = 0 > 2 )- 
These integrals are evaluated in (38) of my paper on ‘ Integrals’; whence I find 
® 3 * = 
where D — ± ^/(l —/c /2 +4/c' 4 ), with upper sign for s = 0 and lower for s = 2. 
It will, however, suffice if we use the development of D in powers of k 2 . The 
result is, in fact, given in the equations next below (38) in ‘ Integrals,’ and they are 
= (!)*( 1-2k»+^k'‘...) > 1 
(56). 
From (53), (54), (55), and (56) we now find 
P-sjl) 03(4) 
W(l)Ca 2 (^) 
e 3 2 
1 _^'2 , _9_ n 
2 5 1 K ' 16 /c • • • 
4 l-2K /2 + f|/c /4 ... 
i 5 i + o^-iW 4 . 
4 1 -k ' 2 + Uk'\. 
— 2_5 
— 4 
(1 + k' 2 -!*' 4 ...), 
-^(l + «' 2 -t«' 4 
(57). 
Thus from (51), (52), (57), we find 
(1+k'4k ,4 ...)^(i+W f 
r 4 \~ 9 /cr 2 '’ 
By (47) f 3 * is equal to the above expressions divided by 
k/* 2 
(58). 
or, a i ^ cos 2 y[ 3(3 -iV) P 
^ ' Al “ 3X? L 1 + “14^ ? ■ • • J • 
It therefore remains to determine & 3 S and Afi. 
