704] 
A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS. 
467 
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 
the second term is here 
= — iri (my + nB) — iri (ay + /38), 
where, my + nB being an even integer, the part — iti (my + nB) does not alter the value 
of the exponential: the effect of the remaining part — nri (ay + /38) is to affect each 
term of the series with the factor exp. — iri (ay + /38), or what is the same thing, 
exp. 7ri (ay + /38), each of these being, in fact, = (—) a y+0 5 . 
We have thus 
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 2z, 2w respectively, 
we have 
which by a preceding formula is 
59—2