470 
A MEMOIR ON THE SINGLE AND DOUBLE THETA-EUNCTIONS. 
[704 
and in the second, writing m', n' instead of m, n, the argument is 
/to' + a , n' + /3' \ 
\ u — u' + y', v — v' + 8'/ ’ 
hence in the product, the argument of the exponential is the sum of these two 
functions, viz. 
= ^ (a, h, b^m + a, n + /3) 2 + ^iri (m + a .u + u' + 7 + n + /3 . v + v' + 8 ) 
+ i («> h, b'^m! + a', n' + /S') 2 + \nri (to' + a'. u — u + 7' + n' + /3'. v — v + 8'). 
Comparing herewith the sum of the two functions 
/> + \ (a + a'), v + | (/3 + fi')\ />' + 2 (« “ °0> v ' + \ (ft ~ P)\ 
[274 + 7 + 7' , 2t; + 8+S' / \2u' + 7 — 7' , 2v' + 8 — 8' / 
= \ (2a, 2h, 2b\[x + § (a + a'), v + ^ (/3 + /S')) 2 
+ ^ 7ri [/x + (cl + a'). 2u + 7 3“ Y 3" v + ^ (/3 + /3). 2v + 8 + 8 j 
3- i (2a, 2A, 26$/ + i(a-af), ✓ + *(£-/3')) 2 
+ ^7rf {/¿' + ^ (a — a'). 2w' + 7 — 7'3-1/ 3- | (/8 — /3'). 2v' + 8 — 8'], 
the two sums are identical if only 
m 4- to' = 2/a, 71 + 71' = 2v, 
to — to' = 2/¿', n — n = 2v, 
as may easily be verified by comparing the quadric and the linear terms separately. 
The product of the two theta-functions is thus 
= 1 exp. 
/> 3- 2 ( a + a ')> v 3- 2 (ft + j3')\ ~ /fi + % (a — a'), v + I {$ — /3')\ 
\2w + 7 + 7' , 2t; + 8 + 8' ) * V2m' + 7- 7' , 2t/ + 8-8' j ’ 
with the proper conditions as to the values of /x, v and of ¡x, v in the two sums 
respectively. As to this, observe that to, to' are even integers; say for a moment 
that they are similar when they are both =0 or both = 2 (mod 4), but dissimilar 
when they are one of them =0 and the other of them =2 (mod 4); and the like 
as regards n, n. Hence if to, to' are similar, ¡x y ¡x are both of them even; but if 
to, to' are dissimilar, then ¡x, fx' are both of them odd. And so if n, n' are similar, 
v, v are both of them even ; but if n, n are dissimilar, then v, v are both odd. 
14. There are four cases : 
to, to' similar, n, n similar, 
to, to' dissimilar, n, n similar, 
to, to' similar, n, n' dissimilar, 
to, to' dissimilar, n, n dissimilar. 
In the first of these, fx, v, fx', v are all of them even, and the product is 
=® (* ( “+?; * ( f+V) (2m - 2s) • e C i - 7. h ^ : 8°) (2m ■ 2!/) -