1895.] M?^ Baker, On a certain auto^norphic function. 325 



These results are obvious by inspection, and the ordinary Riemann 

 process of considering the contour integral 



2^./x (?,»)<<? 



Denoting now by v one of the p every-where-finite functions, 

 the quotient 



(i) is automorphic, since the factor [X{^n, ^0/^(^> *^)P i^ ^^^ 

 same as d^/d^n ', 



(ii) vanishes, within the space S, to the second order at the 

 p zeros, ??2i , . . . , nip, of X (^, m) ; 



(iii) is infinite, within the space S, Sit ^ = m (which we suppose 

 within S), to the second order, and at the zeros of -yr , which are 

 known to be 2p — 2 in number. Denote these zeros by 



Hence, by Abel's theorem, denoting by mi^ the repetition of the 

 place mi, the places nii^, mi, ..., nip- are coresidual with the places 



Consider next, within the space ^S', the function 

 X{^,m)E{i:,m). 



By the properties already enunciated this function has no poles : 

 and has zeros at mj, ..., lUp. It satisfies the equation 



\{tm)E{i:, m) ^ ^ " 

 It is known that, corresponding to any place m, p places 

 Hj , . . . , np 

 can be chosen, such that the function 



% {vf ^ - t»/i' '^' - ... - Vi^p' *V) 



has ^ = Zi, ..., ^p for its zeros. The places Wi, ..., np are definite ; 

 they are such that Wj-, . . . , m^- are coresidual with 



Hence there is an equation of the form 



