449 
(28), 
(29) , 
(30) 
j 
(31) . 
Hence at ( r , 0 , z) of this surface we have, from P= , of (6) 
above, 
T'2 
p = — (r- a) 
r 
= - 2( r=u ) cos mz cos (nt - id) . . (32). 
|a d n-iw 
Hence, and by (6), and (26), and (25), and (23), the condition 
P + p = 0 at the free boundary gives 
4 [Ci; (ma) + «i; (ms)] + ~ tc> )\ ci ; (ma) + dl^rns)] = 0 (33). 
H m 
Eliminating C/C from this by (27) we get an equation to determine 
n, by which we find 
n = u>(i±J N) . . . (34), 
where N is an essentially positive numeric. 
of Edinburgh, Session 1879-80. 
Hence to a first approximation, 
0 = 
ct 
and therefore, by (6) 
whence 
r = g cos mz sm I n — ^ 
r — r 0 £-?- cos mz cos (nt — ^0) . 
n - 
r 
>>2 
Hence the equation of the free boundary is 
r = a — (r=a) cos m 2 cos (nt - id) 
n-iu) 
where 
II. — Sub-case. 
A very interesting Sub-case is that of a = oo , which, by (27), makes 
C = 0 ; and therefore, by (33), gives 
N = ma 
- l'(mn) 
l(ma) 
(35). 
