820 
MR. A. R. FORS5TTH ON THE THETA-FUNCTIONS, 
and 
Ilenco 
Also 
dH_ (2m + yu,) 3 7r 3 
dx*~ 4K 3 U 
du _ (2m+/i) 3 ^ iirx{2m + p) dK 
dp~ 4 p U ~ 2K 3 ~cty U 
du _ inrx{2m + p) 
X dx 2K 
du_ dhi _ldK du 
dp ir % p dx 2 K dp X dx 
K' 
p — 6 "‘K 
dW dK 
1 dp_ ^ d K ^ d K 
' ' p d K ~~ 7T K 3 
(107) 
= “^ 3 ^- ke, - k,e + kk 1 
2/e/e /3 K 3 
Multiplying (107) throughout by ^ and substituting m the first term on the right- 
hand side the value just found for ~-y, we have 
jo ctrc 
du _ 1 d 2 u 1 dK du 
dk~~ 2«7c /3 ’^“K ~cU X dx 
and hence <E>, the sum of the terms u, satisfies 
d 3 c|) 
~chd 
2 kk'* dK d<t> , /0 d<f> n 
-x— + 2/c/c' 2 —=0 
K dK dx 
dK 
which is (102). The quantity — may be explicitly expressed in the terms of p as 
follows. We have 
— p 2 .l — pM 
—p Cl . . . (1 + 2 \p cos 2a:+29 3 )(l+2 p z cos 2cc+p 6 ) . . . 
and 
