230 PROFESSOR M. J. M. HILL ON A SPHERICAL VORTEX. 
From (LXIV.) it follows that 
V {z — ^) — .(LX^.) 
Substituting in (LXIII.) 
Therefore 
Hence 
3Z , dr 
^ ” v/(L + rV _ ,.4) 
(LXYI.) 
dr 
\ v/(L + - H) 2a2 
di 
3Z 
— —o = constant 
H- 
= C 
3Z 
J y/(L + ?-Vt“ — ?’b ' 
. . (LXVIL). 
. . (LXYIIL), 
where, after the integration is performed, L must be replaced by r~ {If- — a-]. 
To deteimine C, it is necessary to substitute in the equation 
0T oia _Oa, d/j, Cfj. 
dr 
dz 
dr dr dz 
. . . (LXIX.), 
i.e., 
3Z _ Z \2 „ 
•2cd ^' “ mb 
z, - ^ - -) + 
- { 21 ' (U= - a?) + 2,'=} - Cr- (z - Z) [ 
dr 
I (L + r~a~ — r*) 
3 2 
Therefore 
Therefore 
Hence 
Hence 
'iif' ' ' 'n." 
LOT 
0=5. 
_ . r_ ± 
~ ^ ]fo(L+ A 
15Z 
j .) j , 
ar — /■') 
. . (LXX.). 
^1 = 1^ '■= L. 1 z, - 
. (LXXL). 
x|^zz: -«')[- r^z - Z) [ 
dr 
1(L + r-a- - rb3/2 
(LXXII.) 
Therefore 
