MR. A. R. FORSYTH ON THE THETA-FUNCTIONS, 
816 
that is 
Similarly 
f 4K 4A 1 27rix 
<f> j 03+ log p, y+ r log r v =p~ % e~ K <&{*, y}. 
4-K 4A 1 2wi!/ 
<3E>l(r+—rlogr, ?/+— r log q [■ = q~ i 'e~ A &{x, y) ! 
7TZ 7 T% \ 
(■•••• (96)- 
which give the two pairs of conjugate quasi-periods, 
4K , , 4A, 4K _ ,4A, 
v log p and — log r, — r log r and —; log q. 
£ & L 7 n & ’ irt & iri ° 1 
TVl 
28. The verification of the expansions in doubly infinite series of sines and cosines 
is easily effected : for substituting in (92) the expressions 
/I / \ ~ 7TX a 27 rx , _ Q 37 TX , 
@o, o( x ) = 1 + 2y> cos —+2p 4 cos + 2p 9 cos — + . . 
0o,o('!/) = 1 + 2 d GOS 7 ^ / +2q 4 cos^+ 2 q y cos^+ . . . 
K 
27 ry 
K 
37 ry 
A 
A 
A 
the coefficient, on the right hand side, of cos is 
r a , , , , (2 log r) 3 „ „ „ , 
p m cf — 2 mn log r.p m q n '-\ - , m~n 2 p m ~(f — . . . 
= 2 
—» 2 MU 
which is right. 
Second proof of the product theorem (23). 
29. The product theorem for single theta-functions, as given by Professor Smith 
(Lond. Math. Soc. Proc., vol. i.) is, with the notation of Section I, 
2 n^ i ( £C )=n^ i (X)+n^, A+1 (X)+(—xr'{n^ +Ifi (X)—n^ +lii+1 (X)} . (97) 
Now, using (94), we have 
®{^ P v) X ’ y> \ =G ^w) 0^fx)0^ A (x')0 Vtp (y)0 Ap (f) 
and therefore 
f, fX n\ 1 2KA log )•/ 0 (P 0 d* \ „ . 
n ®{\ v) Xj V J =e ~ 7r~ \dX\dyp dx^dyp dx^dyp dx^dyJYl. 0^ fX]X\u v p (?/) • * (98) 
