476 MATHEMATICS 



Another line of investigation relates to the work of Poincare", 

 Borel, Fade", et al., on divergent series. It is, indeed, a strange vicissi- 

 tude of our science that these series which early in the century 

 were supposed to be banished once and for all from rigorous mathe- 

 matics should at its close be knocking at the door for readmission. 



Let us finally note an important series of memoirs on integral 

 transcendental functions, beginning with Weierstrass, Laguerre, and 

 Poincare*. 



Algebraic Functions and their Integrals 



A branch of the theory of functions has been developed to such 

 an extent that it may be regarded as an independent theory; we 

 mean the theory of algebraic functions and their integrals. The 

 brilliant discoveries of Abel and Jacobi in the elliptic functions from 

 1824 to 1829 prepared the way for a similar treatment of the hyper- 

 elliptic case. Here a difficulty of gravest nature was met. The cor- 

 responding integrals have 2p linearly independent periods; but as 

 Jacobi had shown, a one valued function having more than two 

 periods admits a period as small as we choose. It therefore looked 

 as if the elliptic functions admitted no further generalization. 

 Guided by Abel's theorem, Jacobi at last discovered the solution to 

 the difficulty (1832); to get functions analogous to the elliptic func- 

 tions we must consider functions not of one but of p independent 

 variables, namely, the p independent integrals of the first species. 

 The great problem now before mathematicians, known as Jacobi 's 

 Problem of Inversion, was to extend this apergu to the case of any 

 algebraic configuration and develop the consequences. The first to 

 take up this immense task were Weierstrass and Riemann, whose 

 results belong to the most brilliant achievements of the century. 

 Among the important notions hereby introduced we note the fol- 

 lowing: the birational transformation, rank of an algebraic con- 

 figuration, class invariants, prime functions, the theta and multiply 

 periodic functions in several variables. Of great importance is 

 Riemann 's method of proving existence theorems, as also his repre- 

 sentation of algebraic functions by means of integrals of the second 

 species. 



A new direction was given to research in this field by Clebsch, who 

 considered the fundamental algebraic configuration as defining a 

 curve. His aim was to bring about a union of Riemann 's ideas and 

 the theory of algebraic curves for their mutual benefit. Clebsch's 

 labors were continued by Brill and Nother; in their work the tran- 

 scendental methods of Riemann are placed quite in the background. 

 More recently Klein and his school have sought to unite the tran- 

 scendental methods of Riemann with the geometric direction begun 

 by Clebsch, making systematic use of homogeneous coordinates and 



