PROFESSOR M. J. M. HILL OH A SPHERICAL VORTEX. 
231 
0r 
— r — X 
3Z 
0/LA 
0r 
= ^,r{z-Z) 
-a^) 
1 _ 
r {z — Z) 
— [r(R'^ — a®) + 
s,\ 
dr 
(L + r^a- — ■?’*■) 
4\3/2 
0~ 
^ = ?t' - X 
0/A 
= X {5«-— 3 (^ - Z)= - f,r'j + II (R^ - a=) (2 - Z) f 
rf?’ 
j (L + 7’”rr — 7’') 
•i!2 
Next, 0x/S^ can be found by means of (LXI.) 
^ _ 
dt 
p_ 
p 
z. 
p 
Z 
p 
- (f- + ^) + 2 (^' + ^ ^^ ^ 
dr dz 
z 
dr 
dz 
+ V)-i(r= + ,.*) + X(r|;;+»| 
-(7 + v)-*(r^+w=)-X®'‘ 
0^ 
n _ 9 z^ 
p 8(d 
— Z'rZ d —7 -(z — Z)"^ + 2a'^ (z — Z)^ 
Z^ 
- Q , [25a^ - 300.2 (z - Zf - GOr^o^ + 9 (z - Z)^+ 36 (z 
37 
L|_5Zr72-Z)j 
d?’ 
(L + 7'2a2 _ 
therefore 
0< 
R 
p 
8fd 
100 
— 24?’2 — 12 (z — Z)2 
15ZZ^ (R2 - rt2) (z - Z) r dr 
4(d J (L + rhd - 
(LXXIIT.), 
(LXXTX^) 
Z)27’'-^+ 3Gr"J 
. (LXXV.) 
Next taking U as the potential of the distribution of matter inside the sphere 
which would produce the jiotential of the irrotational motion outside the sphere 
