FIGURE AND STABILITY OF A LIQUID SATELLITE. 
213 
By means of these transformations we find for X = 1 
f(a\ _3_ 
J \rj 16 1/3 
—§- + i//a + ( — 74 + 47 cos 2 y + 67 cos 2 (3) 
a c 
30 r 
+ ( — 272 + 300 cos 2 y + 356 cos 2 /3 — 123 cos 4 y —151 cos 4 (3 
7 Or 
— 110 cos 2 /3 cos 2 y) 
(46). 
a • • 
If we put (3 = y = 0, and note that c becomes , this expression reduces as 
Zi 
before to the correct form. 
§ 14. Ellipsoidal Harmonic Deformations of the Third Order. 
The two ellipsoids whose forms have been determined are subject to further 
deformation by harmonic inequalities. The expression for the ellipticity f s 
corresponding to all the cosine-harmonic functions for i greater than 2 and s even 
is given in (25), viz. :— 
_ 
fi= -i(2»+l) 
X®/ 
k 3 cos 2 y 
3\r 3 sin 2 yd 
• 3(3 + + l 
14k 2 r 2 ' 
■ +)• 
I shall begin by considering the ellipticities f and f 2 corresponding to i = 3 and 
s = 0, 2. 
I define 
% s (v) = v( K 2 v-q 2 ), (s = 0, 2).(48), 
where q 2 — f [1 + k 2 + </(\ — |k 2 +k 4 )], with upper sign for 5 = 0 and lower for s = 2. 
Then 
r CO 7 
(?) = W M j ^ • 
Since v is always greater than unity, the function under the integral sign may be 
expanded in powers of l/c, as in (7), (8), (9) of § 5, and the integration may then be 
effected. In this way I find 
fjjw/x 1 J 1 7 (l + k 2 ) + 10 q“ 7 [9 (3/c 1 + 2 k 2 + 3) + 28 cp (k 2 + 1)+ 40y 4 j 
\? ) - 2 4 1 ^ ' Q n 2 2 4 Q (, II 4 4 
7k ^ I 2.9k w 8.9.11 k ^ 
+ ... 
(49). 
In all the cases we have to consider k 2 is nearly unity and k ' 2 is small. Then, 
since q 2 = +[4 — 2k /2 + ^/(l — k /2 + 4k /4 )], and since the function under the square root 
may be expanded in powers of k 2 , we may obtain approximate expressions for q 2 in 
the two cases 5 = 0, s = 2. 
When 5=0, we have 
