PROFESSOR M. J. iVl. HILL ON A SPHERICAL VORTEX. 
225 
This leads again to (XLII.). 
The equations (XXX.) and (XXXVIL) must give the same value fov pjp + V. 
This requires that 
Z = 0, 
2F = f Z^ 
and 
T = f Z -^ + J F + S-. 
P 
The first and second of these follow from (XLII.). 
The last gives 
T = |f Z^d- 
n 
Hence (XXXIY.) can be written 
+ V = [3 (g _ Zf - R3]/(2E5) 
- a^Z^ [3 {z - Zf + E3]/(8E8) 
+ IIZ3+ 
P 
Therefore 
^+v=iz2r|5 
+ 3cos2 6» 
(XLIIL). 
Hence at the surface of the sphere 
+V = iZ“(9cos2f)+|) + T 
P P 
(XLIV.). 
Further, outside the sphere E > a, therefore, 
therefore, 
> 0, 
ib 4- V > ^ 
P P 
Now using the value ^ = | Z from (XLII.), putting Z = 0, equations (XXIII.) and 
(XXV.) give inside the sphere 
MDCCCXCIV.—A. 2 G 
