PHILOSOPHICAL TRAx^SACTIONS. 
I. On the Co7inexion of Algebraic Functions ivith Automorpliic Fimctions. 
By E. T. Whittaker, B.A., Fellow of Trinity College, Cambridge. 
Communicated by Professor A. R. Forsyth, Sc.D., F.P.S. 
Received April 23,—Read May 12, 1898. 
It is well known that if 
§ 1. Fntrod'uction. 
f{u, «) = 0 . 
( 1 ) 
is the equation of an algebraic curve of genus {genre, Ceschlecht) zero, then u and 2 
can be expressed as rational functions of a single variable t. If, however, the genus 
of the curve (1) is unity, u and 2 can be expressed as uniform elliptic functions of a 
variable t. 
The natural extension of these results was effected in 1881 by the discovery of 
automorphic functions; whatever be the genus of the curve (1), u and 2 ; can be 
expressed as uniform automorphic functions of a new variable. 
This result is of great importance in the study of algebraic functions. Instead of 
taking z as the independent variable, and studying functions of « on the Riemann 
surface corresponding to the equation (1), we can take t as the independent variable, 
and consider the functions in the plane of L We thus avoid the multiformity of the 
problem, and can apply the simpler and more developed theory of uniform functions. 
Comparatively little of the published work on automorphic functions, however, has 
been written in connexion with the uniformisation of algebraic forms ; in describing 
either groups applicable for the purpose, or the analytical connexions which exist 
between u, z, and t. The only automorphic functions known hitherto which have 
been applied to uniformise forms whose genus is greater than unity, are those given 
by certain sub-groups of the modular group (which will only uniformise special 
curves, containing no arbitrary constants), and those in vi^hich the fundamental 
polygon is the space outside a number of non-intersecting circles. These latter have 
VOL. CXCII.—A. 
2.12.98. 
B 
