861] A FORMULA RELATING TO THE ZERO-VALUE OF A THETA-FUNCTION. 443 
and the formula thus is 
MO) 
V- 
(1 + *) 
But observe that, in the theta-function as defined by the equation 
(x) = % e am2+ - mx , 
a is used to denote the value 
a = a'B, = ^ B, 
A 
where A is the above-mentioned integral, and B is the integral 
rk * dx f 1 k dx 
B 
-f, 
k 2 x — k x . x — Jc 2 . x — k 3 . x — k 4 ’ J -i V1 — x 2 .1 — №x 2 
= 2kK, 
which value must however be taken negatively, viz. we must write B = — 2lcK, and 
we then have 
2ttK 
viz. writing as usual 
a = 
■kK' 
K ’ 
_ttK 
(j — g A ^ y. — q K' , 
the e a of the theta-function is not — q, but it is = r 2 ; and the zero-value MO) is 
= 1 + 2r 2 + 2r s + 2r 18 + .... The equation thus is 
1 + 2r 2 + 2 r 8 + 2 r 18 + 
-V- 
(1+*) 
which is right; in fact, writing A/ in place of k, and consequently K, q in place of 
IF, r respectively, the equation becomes 
1 + 2(f + 2q 8 + 2q ls + ... = ¡J K -Q±h) ■ 
we have 
1 + 2 q + 2q 4 + 2q 9 + ... = \J^~r» 
^ (1 + A;') 
and changing into <? 2 , then (Fund. Nova, p. 92) K is changed into — , and 
we have the formula in question. As a verification for small values of q, observe 
that we have 
2K _ , . 1 + A? _ 
— = 1 + 4>q + 4q-, —■=— = 1 - 4c/ + 16q-, 
7r 4 
and thence 
= or = 1 + ^ 
Cambridge, 12 February, 1886. 
56—2