FIGURE AND STABILITY OF A LIQUID SATELLITE. 
225 
and thence of v 0 , v u v 2 ... in series proceeding by powers of g 2 ; and thence we may 
find k' 2 in that form. 
The result as far as g i is 
12 t + ^ 
K 
4 + X 
1- 
30 2 12 (11+ 26X) 
7{l+X ) 9 7 2 (1 +X) 2 
I have also found the term in g 6 , but shall make no use of it. 
With this value of k 2 or of k 2 , which is 1 — k! 2 , we develop the expression for a 3 //' 3 
in the same manner. The result is :— 
a 3 t> 1 T X 2 
r 3 ^ 1 4 + X^’ 
, 5 + X 2 88 +53.X 4 
1 + —— 
7 (4 + Xy 7 2 (4 + X) 2 
By inversion we have 
2 r * 2 _ 5 (4 + X) a 3 25 (5 + X) (4 + X) a 6 125 (69 + 2X) (2 + X) (4 + X) a 9 
g _ sin y 2(1 + x)r 3 2 2 .7 (1 +X) 2 r ti 2 3 .7 2 (1 + X) 3 r 9 ”" 
a 
Since -- CQS . ^ ^° S y = —a 3 , it follows that « v 
sm 3 (3 1 + X sm 2 (3 \ 1 + X / (cos (3 cos y) 
The semi-axes of the ellipsoid are given by 
a 
X 
2/3 
a 2 c 2 9 
6 2 c 2 
a 2 sin 2 (3 \ 1 + X/ (cos (3 cos y) 2 - 3 ’ a 2 a 
2 v' o 9 n 
9 cos y, — = — cos" U. 
a" a" 
But cos 2 y = 1— i/ 2 , cos 2 (3 = 1 —K 2 g 2 , and therefore 
..2 / \ \ 2/3 2 / \ \ 2/3 
\ /i 9\ —1/3 t-* 9 9\ —1/5 
6+/ X 
a 2 V 
. 1 + X 
1 + X, 
(! ~g 2 )~ is (1 —K 2 g 2 ) 23 . 
Setting apa+t the factor [X/(l+X)] 2 ' 3 , which is common to all, these three are all 
expressible in the form, say 
ft — 1+ (tto + (+«■") g“ + (6 0 + &ik 2 + boK 1 }g l + (co + c^K' + c^K^ + Cg/f 1 ) c/ 6 + ... , 
where the a 0 , a u b 0 , b it &c., have different numerical values according to whichever of 
the three functions we are treating. 
c5 
Now the above formula for k ' 2 enables us to write 
= A 0 -\-B 0 g 2 + Cog *... , 
where the forms of A 0 , B 0 , C 0 are obvious. 
Hence we have 
ft ~ 1 U (<+>UctiH 0 ) g 2 + (£»,, +o + b 2 A 0 2 j g l 
+ ( c 0 + c x A 0 + c 2 A 0 2 -j-c 3 A 0 3 + biBij + 2 b 2 A u B 0 + a x C 0 ) p G ... 
2 Gr 
VOL. CCYI.—A. 
