1895.] Mr Baker, On a certain automorphic function. 323 



within Gi ; then fii is a real quantity less than unity. This trans- 

 formation may also be written in the form 



r=^?|^\=^.(r)say, 



where ( ^.^ ^. ) 



= ( {Bifir^ - Aifi^)l{Bi - Ai), - AiBi ifif-' - f^f)KBi - At) ) 



|( nr^- ,xA)l{Bi-Ai), -{AifjLr^-Bif,i)l{Bi-Ai)\' 



so that ttiSi — fii'yi = 1. The sign of yu,^* is immaterial so far as the 

 transformation Si is concerned ; but it is of importance when we 

 define functions in terms of the quantities a^, /S^, ji, 8i. Hence we 

 define /ii^ = (— )^*~^ I /"■i" I' ^^i being or 1, and at our disposal. 



By successive application of the p fundamental transformations, 

 and their inverses, we obtain a group of substitutions. We may 

 use '^i to denote the general substitution of the group, and write 

 ^^ = ^^- (^). The p fundamental substitutions may be distinguished 

 by the suffix n. Then ^^ is a product of positive and negative 

 powers of the p fundamental substitutions ^„. An automorphic 

 function of ^, in the narrowest sense, is a function which is 

 unaltered when ^ is replaced by any of the transformations ^j. 

 In a more general sense, we consider also, under that name, 

 functions which are multiplied by a definite factor when ^ is re- 

 placed by ^i. Of such, a function of great importance is that given 

 by the definition 



wherein the product extends to every substitution of the group 

 except the identical substitution and the substitutions which are 

 inverse to others occurring. 



This function is infinite only at the place ^ = oo . It vanishes 

 at f = 7 and at all the places ^j (7). The function has an essential 

 singularity at the singular points of the group. Except for the 

 places named the function is everywhere finite and different from 

 zero. 



Further the function has the property expressed by the equation 



wherein %n is one of the p fundamental substitutions, v,/' ^ is one 

 of ^ every- where-finite functions of definite character, Tnn is one of 



24—2 



