468 
A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS. 
[704 
or the function is altered at most in its sign. And again writing 2z, 2w for z, w, 
we have 
^ Cy $) + V + = ^ (7 s) ( U ’ V ^‘ 
In reference to the foregoing results we say that the theta-functions have the 
quarter-periods (1, 1), the half-periods (2, 2), and the whole periods (4, 4). 
The conjoint quarter quasi-periods. 
10. Taking x, y integers, we consider the effect of the change of a, v into 
u + i (ax + hy), v + ~.(hx + by). 
7T% 7T% 
It is convenient to start from the function 
h~ !> ){ u + li ( ' lucJrhy) ’ v + h(hx + by)); 
the argument of the exponential is here 
(a, h, b^m + a — x, n + /3 — y) 2 
+ \in jm 4- a — x. u + 7 + {ax + hy) + n + /3 — y . v + 8 + (hx + by)|, 
which is 
= 5 {a, h, b$m + a, n + /3) 2 + ^iri (m 4- a . u + 7 + n + (3 . v + 8) 
+ other terms which are as follows : viz. they are 
— I {a, h, bjjm + a, n + /3J[x, y) + \ (m + a . ax + hy + n + /3. hx + by) 
+ l {a, h, bjjx, y) 2 - 1rri (x. u + 7 + y . v + 8) 
— ^{x.ax + hy+y. hx + by), 
where the terms of the right-hand column are, in fact, 
= + \ {a, h, 6$ra + a, n + /3][x, y) 
. — \rri (x. u + 7 + y . v + 8) 
— 2 (a, h, bjjx, yf, 
and the other terms in question thus reduce themselves to 
— J (a, h, b\x, y) 2 — £7ri (x. u + 7 + y. v + 8), 
which are independent of m, n, and they thus affect each term of the series with 
the same exponential factor. The result is 
B~ y ){ u + ^l(<» + %>. v + h(kc + by)) 
= exp. I- J (a, h, b(jx, + y+ >/.» +8)) (a, v);