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 (od const.) 
f = c cos mz sin (nt - i6), where r = a 
r = c cos mz sin (nt - iO), „ r = a 
c, t, m, n, a , a' being any given quantities 
and i any given integer 
The condition T = wr simplifies (9) to 
j 
(n - io>) i ( 
if rmhr 
. Xdw 2 i(o 
n — iw) - - — w 
J dr r 
m{ 4w 2 - (n — iw) 2 } 
/ . . f 0 dw i(n-iw) ) 
(n - iw) < 2o) — - -a L w > 
(dr r ) 
T ~ m{4a) 2 - (n - iw) 2 } 
and the elimination of g and r by these from (8) gives 
d 2 w 1 dw i 2 w » 4w 2 - (n - iw) 2 
J- -J- v / on — 
dr 2 ^ r dr 
(n - i(ji>y 
= 0 
or 
where 
d 2 w , 1 dw i 2 w , o A 
-T-2 + --7- ~ f + '" i '“ () 
dr A r dr r 1 
4a> 2 — (n - iio) 2 
(n — iw) 2 
d 2 w 1 dw i 2 w 9 A 
d? + ?dF~7s*, ,A " m0 
where 
<r=, V 
(n - iw) 2 - 4a> 2 j 
(n - iw) 2 j 
(11) , 
( 12 ) . 
( 13 ), 
( 11 ), 
( 15 ), 
(16). 
Hence if g { 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 $T finite or zero for infinitely great values of r ; and 
if I { and denote the corresponding real functions with v imaginary, 
we have 
W7 = CJ i (vr) + Cl i (vr) . . . (17), 
* Compare Proceedings, March 17, 1879, “Gravitational Oscillations of 
Rotating Water.” Solution II. (Case of Circular Basons). 
