704] A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS. 
465 
even integer values of m, n, instead of (a, h, b][m, n) 2 with even or odd values; and 
I write the other term ri(mu+nv), instead of mu + nv; this comes to affecting the 
arguments u, v with a factor 7ri, so that the quarter-periods (instead of being 7ri) 
are made to be =1. 
2. We write 
and in like manner 
fm + a, n+ /3\ 
V u 4- 7, v + 8/ 
= | (a, h, b^m, nf + \ iri {mu + nv), 
= I (a, h, Kfui + a, n + /3) 2 + \nri {(m + a) (u + 7) + (n + /3) (v + 8)}, 
and prefixing to either of these the functional symbol exp. we have the exponential 
of the function in question, that is, e with the function as an exponent. 
We then write, as the definition of the double theta-functions, 
* 
fm + a, n + /3\ 
V u + 7, v + S j ’ 
where the summation extends to all positive and negative even integer values (zero 
included) of m and n respectively: a, /3, 7, 8 might denote any quantities whatever, 
but for the theta-functions they are regarded as denoting positive or negative integers; 
this being so, it will appear that the only effect of altering each or any of them by 
an even integer is to reverse (it may be) the sign of the function; and the distinct 
functions are consequently the (4 2 =) 16 functions obtained by giving to each of the 
quantities a, /3, 7, 8 the two values 0 and 1 successively. 
3. We thus have the double theta-functions, depending on the parameters (a, li, b) 
which determine the quadric function (a, h, b\m, n) 2 of the disappearing even integers 
'a, £> 
(m, n), and on the two arguments (u, v): in the symbol 
7, 8 
which is called 
the characteristic, the characters a, /3, 7, 8 are each of them =0 or 1; and we thus 
have the 16 functions. 
The parameters (a, h, b) may be real or imaginary, but they must be such that 
reducing each of them to its real part the resulting function (* n) 2 is invariable 
in its sign, and negative for all real values of m and n: this is, in fact, the condition 
for the convergency of the series which give the values of the theta-functions. 
4. The characteristic 
ay + /38 is even or odd. 
is said to be even or odd according as the 
sum 
Allied functions. 
5. As already remarked, the definition of