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



where v, w are real ; and the function of § 4 by 



where t belongs to a point in fi 03 and t is the conjugate complex 

 quantity. 



The function of § 1, which is real on (7/ and C 0) and has only 

 real periods, and has one arbitrary pole of the first order in X2 , is 

 given by 



(P + iQ) V) u l + ^ % + (P - iQ) ^M±^ + 4P 



where P, Q are real quantities. 



5. We consider now the case in which the circles C 0> 0/, . . . G p ' 

 are all cut at right angles by another circle 0. It is known that 

 in this case the functions arising reduce themselves to hyper- 

 elliptic functions. The figure (1) drawn for p = 3, will explain the 

 notation we adopt. 





a vj 



Fig I 



In any case where the 2p circles 0/, G 1} ..., G p ', C p consist of 

 pairs of circles inverse to one another in regard to the real axis G , 



