466 
A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS. 
[704 
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+ ^. But the 
functions thus obtained are not regarded as theta-functions, and the expression theta- 
lunction will consequently not extend to include them. 
Properties of the Theta-Functions: Various sub-headings. 
Even-integer alteration of characters. 
6. If x, y be integers, then m, n having the several even integer values from 
— oo to +oo respectively, it is obvious that m + a + 2x, n + /3 + 2y will have the same 
series of values with m + a, n + /3 respectively; and it thence follows that 
/a + 2x, /3 + 2y 
\y » 8 
) 0> v) = % 
Similarly if z, w are integers, then in the function 
* 
a , ß \ 
y +2 z, 8 + 2 w) 
(u, v) 
the argument of the exponential function contains the term 
n [m + a. u + y + 2z + n + /3. v + 8 + 2w}; 
this differs from its original value by 
I iri (m + a. 2z + n + /3.2w), 
= iri (mz + mu) + 7ri (az + ßw), 
and then, m and n being even integers, mz + mu is also an even integer, and the 
term iri (mz + mu) does not affect the value of the exponential : we thus introduce 
into each term of the series the factor exp. iri (az + /3w), which is, in fact, = (—) aZ +P w ; 
and we consequently have 
* (“+2,; L 2 J <“•»(“; s) ; 
or, uniting the two results, 
K“ + 2l; 8 + 2w) <“’ f)(- *)• 
This sustains the before-mentioned conclusion that the only distinct functions are the 
16 functions obtained by giving to the characters a, 0, y, 8 the values 0 and 1 
respectively. 
Odd-integer alteration of characters. 
7. The effect is obviously to interchange the different functions.