390 Mr Baker, On Abelian Functions in connexion [Jan. 24, 



p finite zeros of any one of the 2 P functions A, (£, a) are (/x 1; ..., fi p ), 

 tuhere fx r is either a,, or c r , according to the convention adopted for 

 the signs of a, (3, 7, $ in the p fundamental substitutions. A direct 

 verification is given below. 



Therefore, from what was proved in the general case, a function 

 which is uniform in the region O, unaltered by the substitutions of 

 the group, real on the circles G , (7/, ..., G p , G lt ..., G p , and in- 

 finite to the first order on each of these circles, is given by 



with the notations explained loc. cit. (k), save that c r replaces b r , 

 this can be put into the various forms following, where /x is an 

 arbitrary place, and a constant factor is disregarded, 



vr (£ c) rs {£, a^ ... ot (£ a p ) 



ot(£ a)nr(C d) ... -or (£, C p ) ' 



_y__ \x-x (aO] ... Q - a? (a p )] 

 a? \ x[x — x (d)] ... [a; — # (c p )] 



In the elliptic case, with 



u 



r. ,* „ w i t — B\a — B 



= 2cOV*> * = — log \ -r / j 



we have, with the convention of sign here adopted, save for a 

 constant factor, 



A, (£, a) _ Vjp (u) — e 2 #>' (w) 



^ (Si c) ~ Vp(M+a>)-e8 ~ f ( u ) ~ e s ' 



the points c 1 , a 1} c, a corresponding to the values e 3 , e 2) e x , 00 of 

 jp (u), respectively. It is easy to verify that this function has the 

 desired behaviour. 



Thus by the function \(£ a)'/\'(£, c) the conformal repre- 

 sentation required in Riemann's paper referred to, is effected. 



6. We explain now how to obtain a direct verification from 

 the series itself of the statements that have been made as to the 

 zeros of the function X (£, a). For this purpose we suppose the 

 circle 0, which cuts the circles G , G{, ... , Cp at right angles, to 

 be the axis of imaginary quantities, the point a being the origin, 

 £ = 0, and the left side of the imaginary axis being the interior of 

 the circle 0. Then we may put 



