810 
MR. A. R. FORSYTH ON THE THETA-FUNCTIONS. 
m=oo co 
*£ Ay>f|^cos,r(f ■ +m+r-^ m ) f=\-M®)-^o&)+l • (65) 
the parameters of 0^ Q (x) being p and K, and of 0 OjO (y) q and A, the same applying to 
the other single theta-functions of x and of y which occur below. 
w*'= 00 rjT-rp tn= 00 ?l=0O 
^i=2 2 p<» + 5 >*cos(m+i )^ ; +2 2 2 p<" + »Y 
m —0 A m =0 71=1 
and hence 
7)1=00 71=00 
jJSm+Dn CQa 
+coa 7?-1 (m +Jr) j, — I 
( 66 ); 
2 2 2 pMr<f 
m=0 n =1 
r ($,M+\)n cog 77- 1 (pi -j_i) | + r~® m+l)n cos n j (m+J) ~ ~ 1 
= ^i @i,o{ x ) (6F). 
71 = oo 771=00 
$ 2=2 2 2 ( " +i) ’cos) (»+i)^r+2 2 2 ^ +1 )-co&5rj~+(«+i)f 
n=0 A «,=0 m= 1 _ L K -Y 
+7-«* +I >” cog it j (m+|)|} 
and hence 
71=00 771=00 
2 2 2 p m2 q( >i+ & s 
7i=0 m=l 
• ( 68 ); 
r {2n+\)m C0 y 7r|“d-(n+i)^ j +7* <3w+]) “ COS ( w _|_ 1)^| 
=-^-Ao (y) (69). 
& 3 = 2 2 2 p^+Wq^+w r ^+i)(n+i) cog ^ j( m _|_iyb_|_^_|_n^ 1 
971 = 0 71 = 0 L LA AJ 
+ a ’" +i,! “ +!l cos it j (m + i) — + i)^ 
}] • • (70). 
Similarly 
7)1=00 71=00 
2 1, 't (—1 )yy cos cos 77 ^— — ^J 
—&i—6a,i( x )—6o,a(y)+ 1 
791=06 71 = 06 f *7* 67 7/1 
2 2 2 ( — l)“ + ^p <w+ * J *g^ r (2wfi)« gin 77 i (m-f 1) -+-/ l 
m=0 71=1 L l A AJ 
(71). 
+f-(»«+i)eiinr|(m+l)g_^} =$ B -0 in (x) , , , (72) 
