442 
[861 
861. 
NOTE ON A FORMULA RELATING TO THE ZERO-VALUE OF A 
THETA-FUNCTION. 
[From Grelle''s Journal der Mathem., t. c. (1887), pp. 87, 88.] 
I had some difficulty in verifying for the case of a single theta-function, a 
formula given in Herr Thomae’s paper “ Beitrag zur Theorie der ^-Functionen,” Grelles 
Journal, vol. lxxi. (1870), pp. 201—222. The formula in question (see p. 216) is 
given as follows : , 
(11) %(0, 0,... 0) = aAU] V Discr. (0, 0,... 0) Discr. (0, 0....0), 
but the denominator factor should I think be (7n) p instead of (27ri) p . Making this 
alteration, then in the case of a single theta-function, p = 1, and the function belongs 
to the radical 
y/x — kj. x — b. x — k s . x — k 4 , 
where 
(Jci, k 2) k 3 , kj = ^ 1, +1, ■+■ ~j^j - 
The determinant |H A (V) | is a single term = A, and the formula becomes 
^ (0) = \ — y/(h - b) (b - k 2 ), 
where k 3 — b, k 4 — b are each = l+-r, and we have therefore 
k 
M0) 
A 
. 1 + 
kb 
also A denotes the integral 
rk,) dx 
k, bx — b. x — k 2 . x — k 3 . x — b 
k dx 
_1 Vl — x 2, . 1 — bx* 
k 
k dx 
1 V1 — x 2 .1 — bx 1 
= ikK',