848 
MR. A. R. FORSYTH OH THE THETA-FUHCTIOHS ; 
Periodicity. 
61. Putting 
iir 
v s =e^ .(223) 
where s has in turn every value from 1 to r, in the definition of <E> there follow at 
once the r distinct sets of actual periods for <E>:— 
x 1 
x 2 
x 3 
X s 
4K, 
0 
0 
i 
0 
0 
0 
4K a 
0 
0 
0 
0 
0 
4K 3 
0 
0 
0 
0 
0 
4K, 
0 
0 
0 
0 
0 
4K, 
and 
\V\, 
• • > , AA , T7 - 
)x lt x 2 , . . . , x s +K s , . . . , x r \ 
. . , Vg, . . . , V r j J 
_ /_J /An A a , . . . , A* +1, . . . , AA 
' ' 1W V a , . ■ . , v s Vr) 
X \i X 2, 
> x s> • • ■ X } 
* {(A A"' ’ ' ■' ’ H*i, .^+2K„ ...,«d 
= (-!)"$ 
{fi- 
A 2 , . , 
■ • > A S) . . 
•»VL 1 
■ ,v s , . . 
L \ v i> 
v a> . . 
• >V r J J 
(224). 
By a method similar to that before used can be obtained the following set of quasi- 
periods :— 
