AND DOUBLE THETA-EUNCTIONS. 
905 
as may easily be verified by comparing the quadric and linear terms separately. The 
product of the two theta-functions is thus 
_y (n + U“ + <*')> v + M& + &)\ ^ // + !(«-«')> z/ + K/3-/30\ 
h \2u + ry + ry' ,2V + 8 + S' ) ' * V \2ri + y-f , 2F + 8-S' )’ 
with the proper conditions as to the values of p, v and of p', v in the two sums respec¬ 
tively. As to this, observe that m, m 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, ri. 
Hence if m, m' are similar p, p' are both of them even; but if m, m are dissimilar 
then p, p' are both of them odd. And so if n, ri are similar, v, v are both of them 
even, but if n, ri are dissimilar then v, v are both odd. 
14. There are four cases 
m, m similar, n, ri similar, 
m, m dissimilar, n, ri similar, 
m, ?ri similar, n, ri dissimilar, 
m, rri dissimilar, n, ri dissimilar, 
and in the first of these p, v, p', v are all of them even, and the product is 
/!(« + «'), l(/3 + yS')\ 
\ 7+y, / 
(2 u, 2v) . © 
\ 7-7 » S —8' j 
(2 ri, 2v'). 
In the second case, writing p+1, p +l for p, p' the new values of p, p' will be both 
even, and we have the like expression with only the characters -^(a — a') each 
increased by 1 ; so in the third case we obtain the like expression with only the 
characters ^(/3-\-/3'), |(/3—/3') each increased by 1 ; and in the fourth case the like 
expression with the four upper characters each increased by 1. The product of the 
two theta-functions is thus equal to the sum of the four products, according to the 
theorem. 
Resume of the ulterior theory of the single functions . 
15. For the single theta-functions the Product-theorem comprises 16 equations, 
and for the double theta-functions, 256 equations : these systems will be given in full 
in the sequel. But attending at present to the single functions, I write down here 
the first four of the 16 equations, viz .: these are 
MDCCCLXXX. 6 A 
