MATHEMATICS IN THE NINETEENTH CENTURY 477 



the invariant theory. Noteworthy, also, is his use of normal curves 

 in (p 1) way space, to represent the given algebraic configuration. 

 Dedekind and Weber, Hensel and Landsberg, have made use of the 

 ideal theory with marked success. Many of the difficulties of the 

 older theory, e. g., the resolution of singularities of the algebraic 

 configuration, are treated with a truly remarkable ease and generality. 

 In the theory of multiply periodic functions and the general S 

 functions we mention, besides Weierstrass, the researches of Prym, 

 Krazer, Frobenius, Poincare, and Wirtinger. 



Automorphic Functions 



Closely connected with the elliptic functions is a class of functions 

 which has come into great prominence in the last quarter of a cen- 

 tury, namely, the elliptic modular and automorphic functions. Let 

 us consider first the modular functions of which the modulus K and 

 the absolute invariant J are the simplest types. 



The transformation theory of Jacobi gave algebraic relations be- 

 tween such functions in endless number. Hermite, Fuchs, Dedekind, 

 and Schwarz are forerunners, but the theory of modular functions as 

 it stands to-day is principally due to Klein and his school. Its goal 

 is briefly stated thus : Determine all sub-groups of the linear group 



a) .- 



where a, ft, y, o are integers and ad fir = 1; determine for each 

 such group associate modular functions and investigate their rela- 

 tion to one another and especially to J. Important features in this 

 theory are the congruence groups of (1); the fundamental polygon 

 belonging to a given sub-group, and its use as substitute for a Rie- 

 mann surface; the principle of reflection over a circle, the modular 

 forms. 



The theory of automorphic functions is due to Klein and Poincare. 

 It is a generalization of the modular functions; the coefficients in 

 (1) being any real or imaginary numbers, with non-vanishing de- 

 terminant, such that the group is discontinuous. Both authors have 

 recourse to non-Euclidean geometry to interpret the substitutions (1). 

 Their manner of showing the existence of functions belonging to 

 a given group is quite different. Poincare by a brilliant stroke of 

 genius actually writes down their arithmetic expressions in terms 

 of his celebrated series. Klein employs the existence methods of 

 Riemann. The relation of automorphic functions to differential 

 equations is studied by Poincare in detail. In particular, he shows that 

 both variables of a linear differential equation with algebraic coeffi- 

 cients can be expressed uniformly by their means. 



