324 Mr Baker, On a certain autoTiiorphic function. [May 27, 



p constants, whose value is defined precisely by a barrier curve 

 drawn to connect the circles Gn, On, and gn is or 1 according to 

 the way in which this barrier is drawn. The product 



is independent of the form of this barrier ; the quantities 



are independent of h^. 



The product which defines the function E (^, 7) is known to be 

 convergent when the circles are suitably situated. More precisely 

 the circles of any pair, (7,/, C„, must be sufficiently small, or their 

 limiting points must be sufficiently distant. Under the same con- 

 ditions the series 



wherein the summation refers to every substitution of the group, 

 is convergent. The object of this note is to prove that its value is 

 given by 



wherein @ {u^ is Riemann's @ series defined by 



M=0O 



(h) (qx-) = SS . . . 'Ze-'^'^ (Uyni+...+u^nj,)+iir (Tii«i2+...+2Ti2n,n2+...)^ 

 n= -00 



There are clearly 2^ functions X (^, m), according to the signs 

 chosen for 



(-)''., (-)^ ..., i-f. 



The function X{^, m) is immediately seen, in the ordinary 

 way, to satisfy the equations 



X(^„, m) = (7„^+S,i)X(^, m), 

 ^{^, mn)={yn'>n + Sn)X{^, m). 



The function X, (^, m) vanishes at ^= 00 , and at p other places 

 outside the 2p circles C/, C,j. We shall denote these by 



mi, Wa, ..., m 



The function is infinite at m and at all the places ^{(m). If 

 the space outside the circles be denoted by S, the function is 

 infinite, within the space '^i(S), at the place which is the analogue 

 of m within that space, and is zero at the p places 



