AND DOUBLE THETA-FUNCTIONS. 
901 
Even and odd functions. 
8. It is clear that — m — a, —n—/3 have precisely the same series of values with 
m-\-a, n-\-/3 respectively : hence considering the function 
the linear term in the argument of the exponential may be taken to be 
which is 
ri{ —m—a. -u-\-y.-\-. — n—/3. —v + S}, 
m— (- a. u-\-y. -f - -'ll -f- S J- — tti £ m a.y. -f- .n -)- f3. S J 5 
the second term is here = — 7ri(myf-?iS) —7rt(ay+/3S), where my+n8 being an even 
integer the part — 7ri(my-\-n§) does not alter the value of the exponential: the effect 
of the remaining part — vfay-j-fiS) is to affect each term of the series with the factor 
exp. — 7ri(ay-J-/68), or what is the same thing, exp. ni(ay-j-j8§), each of these being in 
fact =( — ) ay+/ss . 
We have thus 
viz., 3 
/3 
(u, v ) is an even or odd function of the two arguments (u, v) conjointly, 
'cl, /3\ 
according as the characteristic 
is even or odd. 
The quarter-periods unity. 
9. Taking z and w integers, we have from the definition 
viz., the effect of altering the arguments u, v into u-\-z, v-\-w is simply to interchange 
the functions as shown by this formula. 
If z and w are each of them even, then replacing them by 2 z, 2 w respectively, we 
have 
$ 
CL, ft 
7, 8 
(u+2z,v+2w)=.^ + 2 - Mw 
> ft 
