AND DOUBLE THETA-FUNCTIONS. 
899 
and in like manner 
/m+«, n + /3' 
\w + y, v + S 
\(a, h, bXm + a, n + f3y 2 +lTri{(m3-a)(u+y)(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, 
hn -t- «, n + /3\ 
\m + Y, v + 8 ]’ 
where the summation extends to all positive and negative even integer values 
(zero included) of m and n respectively : a, f3, y, 8 might denote any quantities what¬ 
ever, 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, y, 8 the two values 0 and 1 successively. 
3. We thus have the double theta-functions depending on the parameters (a, h, b) 
which determine the quadric function (a, h, b X m > n Y °f the disappearing even 
integers (m, n) : and on the two arguments (u, v) : in the symbol (*’ ^), which is called 
the characteristic, the characters a, /3, y, 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 ( * X m > n Y 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 
is even or odd. 
is said to be even or odd according as the sum ay-f-/3§ 
Allied functions. 
5. As already remarked, the definition of 
is not restricted to the case where the a, (3, y, 8 represent integers, and there is 
actually occasion to consider functions of this form where they are not integers : in 
particular, a, (3 may be either or each of them of the form, integer fi- <(. But the 
functions thus obtained are not regarded as theta-functions , and the expression theta- 
function will consequently not extend to include them. 
5 z 2 
