446 
Proceedings of the Royal Society 
Crowds of exceedingly interesting cases present themselves. 
Taking one of the simplest to begin : — 
Case I. 
Let T = wr (w const.) 
r = c cos mz sin ( nt - iO), where r = a 
r = t cos mz sin (nt - iO), ,, r — vi ; 
c, t, m, n, a , a' being any given quantities j 
and i any given integer J 
The condition T = ur simplifies (9) to 
(n - toy) 
. \dto 2i(o ) 
n — ioi) - — w > 
J dr r ) 
m{4o> 2 — (n — ii o) 2 } 
, . . f 0 dw i(n-iw) ) 
(n - zoo) < 2o)-~ - — L w > 
{ dr r ) 
m{ 4o> 2 - (n - zo>) 2 } 
and the elimination of g and r by these from (8) gives 
d 2 w Idw i 2 w o 4o) 2 — (n — io )) 2 
+ + m 2 - v ’ — 
dr 2 r dr 
(n - i<*y 
w = 0 
or 
dhv , 1 dw i 2 w l9 A ^ 
+ - — - — + v 2 w = 0 
dr 2 r dr 
where 
/ 4a) 2 - (n - Za>) 2 
= m / ^ 
V (n - ico) 2 
dhv , Idw i 2 w 9 r . 
+ - — cr 2 W = 0 
dr 2 r dr r 2 
where 
=m J 
(n — toy) 2 — 4 w 2 i 
(n - toy) 2 J 
(11) , 
( 12 ) . 
( 14 ), 
( 15 ), 
(16). 
' • ( 13 ), 
Hence if J denote Bessel’s functions of order i, and of the first 
and second kinds,* that is to say J; finite or zero for infinitely small 
values of r , and |f finite or zero for infinitely great values of r ; and 
if I; and denote the corresponding real functions with v imaginary, 
we have • 
w = CJi (vr) + <ftg; (vr) . . . (17), 
* Compare Proceedings, March 17, 1879, “Gravitational Oscillations of 
Rotating Water.” Solution II. (Case of Circular Basons), 
