268 Algebraic Functions and Automorphic Functions. 



dividing the plane into curvilinear polygons corresponding to such 

 a group ; these polygons are simply-connected, and cover completely 

 the half of the plane which is above the real axis. Sub-groups of 

 these groups are found, whose genus is greater than unity, and 

 which are appropriate for the uniformipation of any algebraic 

 curves. 



The sides of the polygons, into which the half -plane is divided, are 

 formed of arcs of circles orthogonal to the real axis. These may, in 

 the sense of Lobatchewski's geometry, be regarded as straight lines. 

 One case, where the construction fails, is shown to correspond to the 

 limiting case in which Lobatchewski's geometry becomes Euclidian 

 geometry; the figure then becomes the division of a plane into 

 parallelograms, used in the theory of doubly periodic functions, and 

 is appropriate for the uniformisation of algebraic curves of genus 

 unity. Thus doubly-periodic functions are a limiting case of the 

 class of functions considered. 



The automorphic functions of the groups described solve the 

 problem of conformally representing a plane, regarded as bounded 

 by a number of finite lines radiating from a point, on a curvilinear 

 polygon, whose sides are derived from each other in pairs by pro- 

 jective substitutions of period two. This leads to the conformal 

 representation of any Riemann surface, at each of whose branch- 

 points only two sheets are connected, on a curvilinear polygon whose 

 sides are derived from each other in pairs by projective substitutions; 

 and, as it is known that any algebraic curve can, by birational trans- 

 formation, be represented on a Biemann surface whose branch- points 

 are all simple, it is seen that the uniformisation of algebraic func- 

 tions of any genus can be effected by groups of the kind described. 



The analytical connexion between the variables of the algebraic 

 form and the uniformising variables is given by a differential 

 equation of the third order. A certain number of the constants in 

 this equation have to be determined by the condition that the group 

 of substitutions associated with the equation leaves unchanged a 

 certain circle. When any arbitrary values are given to these con- 

 stants the solution of the differential equation is termed a qnasi-uni- 

 formising variable. The properties of quasi-uniform ising variables, 

 and their relation to the uniformising variable, are discussed in the 

 last section of the paper. 



