449 
(28), 
(29) , 
(30) 
(31) . " 
/ T 2 dr 
, of (6) 
above, 
T2 
P = — (r- a) 
r 
_ _ c g(r = u) cog cog t . (32). 
|a 3 n - iu 
Hence, and by (6), and (26), and (25), and (23), the condition 
P +p — 0 at the free boundary gives 
4 [Ci; (ma) + «i; (ma)] + (” ~ *' m ) 2 [ci,(ma) + «l ( (ma)] = 0 (33). 
a m 
Eliminating C/C from this by (27) we get an equation to determine 
n , by which we find 
n = <»(i±J N) . . . (34), 
where N is an essentially positive numeric. 
of Edinburgh, Session 1879-80. 
Hence to a first approximation, 
0 = 4 
and therefore, by ( 6 ) 
whence 
= g cos mz sin ^ ' n - t ; 
r = r 0 £- 7 - cos mz cos (nt — iO) . 
ic 
n-—s 
r l 
Hence the equation of the free boundary is 
r — a — (r= f ) cos mz cos (nt - iO) 
n-ioi 
where o> = — h 
II. — Sub-case. 
A very interesting Sub-case is that of a — go , which, by (27), makes 
C = 0 ; and therefore, by (33), gives 
F(ma) 
N = ma- 
i(ma) 
(35). 
