76 
PROFESSOR W. M. HICKS OK VORTEX MOTIOX. 
Case of\ small .—Here y is also small. . If equation (34) be expanded 
y and X, there results 
% 
2^ 2/ -1 ( ) (271 + 3) {2n + I)! 
Dividing by this may be written 
71 “ 
( 27 ^ + 1 )! 
=: 0 . 
X“ 
•7 
+ 30S2(~)'" 
n + 1 
{2n + 3)! 
l(n + l)y-“ - X“], 
whence y can be exj)ressed in terms of X by successive approximation, 
be found that 
y = 
A,“ 
112 
224 
This gives the equatorial axis at 
a 
\/2 
2 2 7 
5 4 
When X = 0, this agrees with Hill’s vortex. 
Case of Xj.—The equation in y for this case becomes 
cosy + y 
y 
sin y = 0, 
y is always nearly = mr — 2 say, where 2 is small. Then 
cos 2 — 7177 — 2 
7177 — Z 
sin 2 = 0. 
Whence 
y = 
mr 
= mr — 
1_ _ _5_ 1f_ 
iiTT 3 {nir^ 15 {mry 
•31831 -05375 -01590 
n 
ir 
IV 
This formula gives for the two lirst roots 
2*75363, 6*11682. 
The values obtained by numerical calculation are 
2*74371, 6*11676. 
in powers of 
To X'’ it will 
(35). 
