822 
MR. A. R. FORSYTH ON THE THETA-FUNCTIONS, 
^l + 2 2 +2^+2g 9 + . . . +2 p |l+^+I)+ 2 *^+^+ 3 ^r 6 +^+ . . . j 
+ V{l+ 2 (r* + g-|- 2 ^ + I) + 2^ + r) + . . . } 
+ 2p^ l+ ? (r 6 +-,)+2 4 (r 12 +i)+ ... f + . . . 
=W {l+^+;)+^+ r l)+ 2 9 (r s +i)+ ■ • • } 
+ ^ l { 1 + 2 ( r 3 + r ’) + 2 '( r 6 + 3 + 2 9 ( r 9 + 3 + • - • } 
+ 2 p v{l+ 2 ^+I)+^io+i)+ g 9 (r«+i^ 4 -...}+... J 
+ _ 22 *{l-?>(r+^+iP 4 (^+^)-i> 9 ^+i)+ . . . } 
+ 2 2 * {l-y(rH^)+^- 6 +^)-^(^+ r i ,)+ • . . } 
+ 29 ^X- i 5 (^+I)+ i ,*(r“+i^_jp»(r“+i)+ ...}+.. . J 
+[l- 22 + 22 *- 22 »+ . . . - 2 i ,{l- 2 ^+J)+ 2 ^+I)- 2 »(r 9 +i)+ . . . } 
+ 2 p*{l-g(^+^+ 2 4 (^+^- 2 9 (^+^)+ • ■ • } 
-V { 1 - 2 (^+ 3 + 2 4 (^+,i)- 2 9 (r 18 +^)+ ...}+... J 
Also equations ( 42 ), ( 43 ), ( 44 ) give expressions for k 1} k 2) k s , in terms of p, q,r; and 
other identities are obtained from the equation (a)-(ie). 
34 . Let k 2 =c, k 2 =c' ; then 6 being any single theta-function 
dW O de d6 
3 -a«£+/**=° 
da? 
where 
, E 2 cc' dK 
a ~ C K K dc ’ 
/3=4 cc\ 
First, let 6 be one of the three even functions : differentiating 2 n times with respect 
to x. and then putting x zero 
d^n+20 &uQ d c p«Q 
^£**0 
dxJ n+ 2 
