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It is well known that iff (u, z ) = 0 . . . . . . . . . (1) is the equation of an algebraic curve of genus (genre, Geschlecht ) zero, thenu andz can be expressed as rational functions of a single variablet . If, however, the genus of the curve (1) is unity,u andz can be expressed as uniform elliptic functions of a variablet . 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 andz can be expressed as uniform automorphic functions of a new variable.

I. On the connexion of algebraic functions with automorphic functions