Algebraic Functions and Automorphic Functions. 267 



In conclusion the author wishes to express his thanks to Professor 

 Ewing for many suggestions as well as for the facilities which 

 have enabled the experiments to be carried out. 



[Subsequent to the writing of the above paper, permeability curves 

 have been taken, for a ring of the same material, by the ballistic 

 method, up to a magnetic force of 25 C.G.S. units. A comparison of 

 those taken before and after baking shows that the saturation value 

 of the induction is unchanged by prolonged heating, for although the 

 earlier part of the curve, as in the case given above, was much 

 altered, the parts of the curves above a force of 15 C.G.S. units are 

 practically indistinguishable. May 15, 1898.] 



44 On the Connexion of Algebraic Functions with Automorphic 

 Functions." By E. T. WHITTAKER, B.A., Fellow of 

 Trinity College, Cambridge. Communicated by Professor 

 A. R. FORSYTE, Sc.D., F.R.S. Received April 23, Read 

 May 12, 1898. 



(Abstract.) 



If u and z are variables connected by an algebraic equation, they 

 are, in general, multiform functions of each other ; the multiformity 

 can be represented by a Riemann surface, to each point of which 

 corresponds a pair of values of u and z. 



Poincare and Klein have proved that a variable t exists, of which 

 u and z are uniform automorphic functions ; the existence-theorem, 

 however, does not connect t analytically with u and z. When the 

 genus (genre, Geschlecht) of the algebraic relation is zero or unity, t 

 can be found by known methods ; the automorphic functions required 

 are rational functions, and doubly periodic functions, in the two cases 

 respectively. But no class of automorphic functions with simply 

 connected fundamental polygons has been known hitherto, which is 

 applicable to the uniformisation of algebraic functions whose genus 

 is greater than unity. 



The present memoir discusses a new class of groups of protective 

 substitutions, such that the functions rational on a Riemann surface 

 of any genus can be expressed as uniform automorphic functions of 

 a group of this class. These groups are sub-groups of groups gene- 

 rated from substitutions of period two. Groups are first considered 

 which can be generated by a number of real substitutions of period 

 two, whose double points are not on the real axis, and whose product 

 in a definite order is the identical substitution. These groups are 

 found to be discontinuous, and of genus zero. A method is given for 



