PROFESSOR W. M. HICKS OK VORTEX MOTION. 
41 
(2) Polar Co-ordinates, {r, 6), 
p — r sin 6 du = dr = dii 
dv — dd 
dn' = r d,9, 
and 
d ( r d-\lr\ , d f 1 d-<Jr\ 
* (7 * ) + M (7 7) = - 
or 
d^yjr 1 d'ylr cot 6 d4r 
^ ^ dO ~ ^ ' ■ 
(3) Spheroids. 
(a) Prolate. Here p zt = \ sinh (ti + vl), 
p — \ sinh u cos v, z — \ cosh ii sin v. 
The surfaces u, v are respectively the ellipses and hyperbolas 
. (7). 
whence 
+ 
sinh^tt cosh^w 
= and 
siii-"y cos^v 
= 
u increases from 0 at the origin to co at an infinite distance ; v increases from — at 
points on the negative part of the axis of 2 , through 0 for points on the equatorial 
plane to |-7r at points on the positive part of the axis of 2 . 
Again 
dn\^ 
du j 
dpV fdzV d , , . d , 
dv) +( 7 ) 
= X' cosh (tf + rC) cosh ~ vl) 
= X“ (coshV — sin"r). 
Hence the differential equation is (writing C and S for cosh sinh u). 
J_ A / i I i 
cos V du\'& du ) S dv \cos v dv j 
+ 7 :^ ( 777: ^ ) = — 47rX^w sin y (C“ — siiffi?) . . (8). 
(yd) Oblate. Here p -f = X cosh [u + vt) 
p \ cosh u cos V, 
z = X sinh sin v, 
= X^ sinh [u + Vl) sinh {ii — vl) 
= X^ (cosh^-M — cos\’), 
and the differential equation is 
d / 1 d-»/r\ , ^ d 
Jr., \ n Jr,, I ri 
cosv du \ G du 
VOL. CXCII. — A. 
1 di\r\ 
C dv \cos'y dv j 
G 
— 47rX®(y sin y (C^ — cos^v) . . (9). 
