PROFESSOR M. J. M. HILL OX A SPHERICAL VORTEX. 
233 
From (LXXXIII.) and (LXXXIV.) it follows that 
dt 
x-U+Ky+ifz^l* 
dz 
x-u + f(7 + Hz^)‘^*' 
Hence x ~ U + + f| dt is a function of r and z — h only, therefore 
3(--Z)L 
X-U+ l(-5 + f|Z")<^* 
= - 5X 
r 
(c-Z)[ 
dr 
IL + 
. (LXXXY.). 
Before proceeding further it is necessary to prove that 
dr 
(L + rhd - 
1 
r'^(L + r^a? — 
^ LyL + r%2 - \ 
L dr 
7’* (L + rhd — 
(LXXXVL). 
Differentiating both sides with regard to r, an identity is obtained. 
Hence the result holds. 
Making use of (LXXXVI.) in (LXXXIL), and remembering that after the 
integrations in (LXXXVI.) are effected, L may be replaced by r® (R^ — a^), 
d_ 
dr 
x-u + |(y+ I|zt* 
= - 5X 
1 _ r (E3 - cd) + 7-^ j ^ 
_?’ {z - Z) (z-Z) 2 0r 1 ’ J 7-^ (L + - 7“^)^'^ I 
dr 
-\r 
Ldr 
(L + 7-2«3 _ ,.1)'!/^ 
J J 
= - 5X 
L 
dr 
^{L + r\d - 7’^) 07’ !_ J 7’i (L + 7’2«2 - ry'^ J 27’^ (L + rhd - r^) 
dr 
0.4\a/- 
15Z 
4ft2 
0L r 
dr 
y yili + i^cd — r*) ^ 0r J27’'^(L + r'^cd — ry^ 0;’ J7’'*(L -(■ r%“ — ry~ 
0L 
dr 
15Z y 
4a^ dr 
u 
dr 
] ,y.4(L _|_ — ry^ 
MDCCCXCIV.—A. 
(LXXXVIL). 
2 H 
