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



The construction of a function, single valued in the region 

 £1, unchanged by the substitutions which transform the circles 

 C{, ... , Gp respectively into G x , ... , C p , and having p + 1 poles of 

 the first order within fl, of arbitrary position, is known — being as 

 follows (loc. cit. (k)) ; let the poles be at z, c 1; ... , c p> and let 



0(2>f) = 2 (M±|)l s ; 



r z r ~ b 



let the additive arbitrary constant be determined by the condition 

 that the function vanishes at £= a ; then, the function is given by 



E <S>(z, £), <D(*, a), <$>(z, a,), ... , <b{z, a p ) 



<S> (c 2 , £), <l> ( Cl , a), 3> ( Cl , a,), ... , ®(c ly a p ) 



3> (c p , £), <£ (c p> a), <£ (c p , aj), . . . , 3> (c p> a p ) 



1 , 1 , 1 , ... , 1 



where E is an arbitrary constant, and a 1} ... , a p are the analogues 

 of a by the p fundamental substitutions whereby the circles 

 Ci, ... , Cp are respectively transformed into C x , ... , C p . In the 

 case under consideration the region Cl has the specialty that the 

 circles C 1> ... , C p are the inverses of C/, ... , C p in regard to C — 

 we proceed to prove that in this case, if z be upon the real axis 

 C , and Ci, ... , c p be respectively upon the circumferences C/, . . . , C p , 

 then, for a suitable determination of the constant E, the function 

 is real upon each of the curves C , C(, ... , C p . The necessary 

 value for E is that of the inverse of the minor of <J> (z, £) in the 

 determinant (or any real multiple of this) — when E is so determined 

 the function may be denotod by -v/r (£, a; z, c 1} ... , c p ). 



To prove this it is sufficient to prove that the function is real 

 upon the real axis (Schwarz, Ges. Abh., II. p. 66). Now it follows 

 from Riemann's theory that the function, uniform in 12, unchanged 

 by the p fundamental substitutions, with poles at z, c lt ... , c p , 

 respectively on G , 0/, ... , G p , and vanishing at £=a, is given by 



->£ a 



/(O = (P, + *&) rt; +... + (p p + iQ p ) rj; + <p + »Q) r 



where P 1} Qi, etc., are real constants whose ratios are given by 



(Pi + iQi) <f>n (Ci) + . . . + (Pp + iQ p ) (j> n (c p ) + (P + iQ) </> n (5) = 0, 



(n = l, 2, ...,^), 

 where <f> n (£) = dv^ a /d%. We have already seen that <f> n (£) is real 

 when £ is real ; hence these equations involve also, if c/, c r be 

 conjugate complexes, the equations, 



(Pi - iQi) <f>n CO + . . . + (Pp - iQ,) <£„ (c/) + (P - iQ) n (*) = 0, 



(71 = 1,2, ...,p), 



