with 
Taking x, y even, or writing 2x, 2y for x, y, then on the right-hand side we have 
which is 
but there is still the exponential factor. 
11. The formulae show that the effect of the change u, v into n H . (ax + hy), 
7Tl 
v -j—(hx + by), where x, y are integers, is to interchange the functions, affecting them 
7Tl 
1 1 
however with an exponential factor; and we hence say that —. (a, h), —(h, b) are 
7Tl 7TI 
conjoint quarter quasi-periods. 
The product-theorem. 
12. We multiply two theta-functions 
Vy f) + U ' } V + V '^ ^ (y g') “ u '> v ~ v ') ’ 
it is found that the result is a sum of four products 
'1(cl-cl)+p, ±((3-/3') + q 
@| i(« + «')+i>> b(P + P + V)(2 u , 2t/"' 1 1(2«', 2v'), 
where p, q have in the four products respectively the values (0, 0), (1, 0), (0, 1), and 
(1, 1); © is written in place of ^ to denote that the parameters (a, li, b) are to 
be changed into (2a, 2h, 26). It is to be noticed that, if a, a are both even or 
both odd, then |(a +«'), | (a - a') are integers; and so, if ¡3, /3' are both even or 
both odd, then £ (/3 + /3'), \ (/3 — /3') are integers; and these conditions being satisfied 
(and in particular they are so if a = a, /3 = /3') then the functions on the right-hand 
side of the equation are theta-functions (with new parameters as already mentioned); 
but if the conditions are not satisfied, then the functions on the right-hand side are 
only allied functions. In the applications of the theorem the functions on the right- 
hand side are eliminated between the different equations, as will appear. 
13. The proof is immediate: in the first of the theta-functions, the argument 
of the exponential is 
im + a , n + /3 \ 
V u + u + y, v + v' + SJ