222 
PROFESSOR M. J. M. HILL ON A SPHERICAL VORTEX. 
and the ellipsoid of revolution, 
If, further, it be supposed that c = a, the ellipsoid becomes the sphere 
r-3 + (2 - Zf = 
(XXIL). 
The discussion will now be limited to this case. 
In it 
ro = Z - 2 4 (2i" - an - 2 4 (z - Zf 
a“ ' ' a- ' 
1 
. (xxiiL) 
5A 
CO =■ 
KV 
ft" 
(XXIV.), 
p 
P +V = f (,-z-0-(z-Z)Z-f (z-Z)* + f (z-Zf+V (XXV.), 
where II/p is an arbitrary function of t. 
i// = 
1 
J 
(XXVI.). 
* The surfaces X = const, are a particular case of some surfaces that were noticed by Professor Lamb 
in a paper “On the Vibrations of an Elastic Sphere,” published in the ‘Proceedings of the London 
Mathematical Society,’ vol. 13, p. 205. 
In equation 75 of that paper, viz., 
i (^0 - f 1 
where 
V'l (•*) — ^ 2 5 2 V V 7 ’ ’ 
the current function may be written 
(Icr) - {ka)} = 
k- 
2.5 
(P 
+ .) 
.4.5.7 
(H 
If we suppose CA:^ to be finite, but k = 0, this becomes 
'iv'- (r'" — a-), 
or, in the notation of this papei', 
CV3 {P + (z- Zy~ - a2}, 
which agrees with the above. 
