232 
PROFESSOR M. J. M. HILL OR A SPHERICAL YORTEX. 
U = Z (2 - Z) [3R2 - 5 a 2 }/( 4 a 2 ) 
by equation (LIl.), therefore 
SU 3Z 
— = „ (2 — Z) 2r 
{j}' 4rr ' ' 
(LXXVL), 
8U _ 3Z 
dz 4fl® 
2 (2 - Z)2 + 112 
5Z 
’4 
(LXXVIL), 
Hence 
- U) _ 
0r 
8U__ 
d( 4a^ 
2 (2 ~ Z)2 + R2 
+ 
3Z2 
= — 5X 
r {z 
(LXXYIIL). 
. (LXXIX.), 
5(x-'b) 
= - 5X 
dr 
j (L + 
(LXXX), 
-J-MZ-+5\Z 
- Z)f: 
dr 
(L + ?’V - ?-q3-3 
(LXXXL), 
therefore 
dr 
X-U+f(" + ||Z^)rf(j 
— - 5X 
r (z - Z) 
_ _ (RS _ „») + ,45 f 
dr 
J(L + rV - 7’b® 2 
(LXXXIL), 
d.r 
X-U + 
" + fl-z47( 
= — 5X 
dr 
+ ,."-^2 - r *) 
•1 \3 ; 
(LXXXTIL), 
d_ 
dt 
A P 
= 5XZ 
X-U + + i|z==)<-;^ 
.5 f(L + rV )■')■' -J 
dr 
1 
(LXXXIY.). 
