826 
MR. A. R. FORSYTH OH THE THETA-FUNCTIONS, 
the zero subscript in the differential coefficient implying that the variables are made 
to vanish after differentiation. Now 
&o=OqA®)0oM 
Remembering that an uneven differential of an even function is an uneven function 
and that an even differential is an even function, we have 
/KA dd M 
^0,0 , 
//KAY 
t r 3 dx 
% 1 
l -* ) 
( l) M N 0)3 , +i 
Let 
d 2n ~*% t o 
d“ s @ o,o i 
l//KA\2fZ2«-^+3(9 0i0 
# s+3 0 oo , 
dx^ n ~ is 
^o 3 * ‘ 
2!\ 7T 2 / 
dr 0 2 * _2 * +2 
dy 0 3 * +3 ‘ 
/KA 
^+^0.0 
1 /r'KA\3#»-^+26l 0i0 
d^e 0,0 
5 9 
7T* 
d(v 0 u ~ Zs 
dy 0 3 * +3 
3i\ 7T 3 y 
‘® +1 
A - 2 ri 4-- ——L— * I 
1 tt|_ ~*~2! dp'dq'' 4:1 dp'Hq'^ 
2 , T .( d? \*1 
=-cosh 
7T 
Ao=- 
7T 
1 + 
1 Ww/ J' 
d? d* 
3! d/dq'^4 dp'Hq'* 
(124), 
=-t— to u - sinh 
7TT 
d a y b |_ \dp'dq 
dp'dq' 
r' 
(125). 
Then we have 
*)•/ 7r\ 2s d 
-IN n. o.e— I ^ 
and 
^0,2*+l— 
c 0 =A 1 .KW.(126), 
.(127), 
i s K*A* .... (128). 
O’ 2 * \K/ \AJ dp'^dq'* 0 ' 
r'KA/7T tj-\ 2 (j+i) d n+ 1 
\K/ \A/ dp hl - s dq w 
The remaining formulae are obtained by an exactly similar process and are as follow. 
r ^'i :=c i yY + • • • 
+^( N W Nl ' 1 ’ •••• N l.» ■ • ■ > ^ s, ‘+.( 12 A 
c 1 =A 1 .K“C i A i .(130), 
where 
^1,2s— 
7T\ 2(«— S) [tt \ 3# d n 
K 
- 
(131), 
\A/ dp' n ~ s dq' s 1 . 
AT /KA/7r\2(«-^)/7r\2(^i) ^+i . ... /1qo x 
and 
