812 
MR. A. R. FORSYTH ON THE THETA-FUNCTIONS, 
W= CO n— oo 
2 i)-+^(-+^(«+i) 2 U2te+i)(«+i) cos ^ j( m +A) A + ( % _|_l)l 
_ r -2(®+i)(»+i) CQS ^ I -{-1) 
=S 15 . . (82). 
From these formula) many other double summations may be deduced : the following 
may be taken as specimens. 
7)1= CD 71= CO 
8 t t p^q 4 ^ 
m=l n=l 
>- cos 2^+^+r-s™ cos 2^f- n £ 
== “h""h"“F^is %@o,o( x ) 2^o,o {y) 2^o,i(*) 2# 0il (y)-{-4 . (83). 
8T T p e».+iY''fr« 2 * +1 >cos tt{ + r -« 2 »+i> cos | (2m+l> _2^r 
w =0 «=1 L L F A J l F A J _ 
^0 ' 9 ’ 4 ,“ b '^'8 ^12 ’ .( 34 ). 
m=cD n=co 
8 t $ p^Y n+iy 
m—1 n—0 
>*«> cos «»,,{?£!. 
(2n4-l)y 
A 
7 ?j = co 71=00 
8 2 A p<^+i) a ^(2«+ir 
)»=0 n—0 
=*0+**-*s-h2- 2^(2/).(85). 
r(2w+ 
l F 
^»+i)(2»+i) cos ^| (2w + l)a) ^ (2?z.+l)y | 
+ ? 
2<2»?+1)(2»+1) cos ff | (2^ + l> (2» + l)y 
—^0—At—-5-8+ ^12 • (S3). 
4.V ”s j*2.* ( 2«+Dcos Tt\~ + {n +|)|1 +,•-«»+«cos nl (n+i)?l 
m=l n =0 L L F AJ [ IV AJ _ 
=5 2 +a 6 -2^, 0 (y).(87). 
m=oo 71=00 
4 ^ £ pttm+Wqiu+i) 2 
m=0 w=0 
r ®»+lX2»+l> coa ^ f 22±1 > + (,j + 1)| 
+J .-8. t te»+lf cog1r j^±i)2_( B+i )|J . (88). 
w = 00 W = 00 
4 2 2 p(M+i)*qW 
m= 0 »=1 
,.8,1(2™+]) COS «•{(«+s)g+“^| +l’ -2 “ <2,,+1> COS 7T | (m + | 
=^+^- 2 ^ 0 ( 0 :).(89). 
it "t p<-+B’ 2 <, “+‘>f^>« + l»>+»coBart(«+i)l+^t^l 
?»=*o w=o L L F A J 
+r -(2*+i)M»+i> oos ir{(m4-i)|-^^}]=^-^ • (90). 
