478 MATHEMATICS 



Differential Equations 



Let us turn now to another great field of mathematical activity, 

 the theory of differential equations. The introduction of the theory 

 of functions has completely revolutionized this subject. At the 

 beginning of the nineteenth century many important results had 

 indeed been established, particularly by Euler and Lagrange; but 

 the methods employed were artificial, and broad comprehensive 

 principles were lacking. By various devices one tried to express 

 the solution in terms of the elementary functions and quadratures 

 a vain attempt ; for as we know now, the goal they strove so 

 laboriously to reach was in general unattainable. 



A new epoch began with Cauchy, who by means of his new theory 

 of functions first rigorously established the existence of the solution 

 of certain classes of equations in the vicinity of regular points. He 

 also showed that many of the properties of the elliptic functions 

 might be deduced directly from their differential equations. Ere 

 long, the problem of integrating a differential equation changed 

 its base. Instead of seeking to express its solution in terms of the 

 elementary functions and quadratures, one asked what is the nature 

 of the functions defined by a given equation. To answer this ques- 

 tion we must first know what are the singular points of the integral 

 function and how does it behave in their vicinity. The number of 

 memoirs on this fundamental and often difficult question is enormous; 

 but this is not strange if we consider the great variety of interesting 

 and important classes of equations which have to be studied. 



One of the first to open up this new path was Fuchs, whose classic 

 memoirs (1866-68) gave the theory -of linear differential equations 

 its birth. These equations enjoy a property which renders them 

 particularly accessible, namely, the absence of movable singular 

 points. They may, however, possess points of indetermination, to 

 use Fuchs 's terminology, and little progress has been made in this 

 case. Noteworthy in this connection is the introduction by v. Koch 

 of infinite determinants, first considered by our distinguished coun- 

 tryman Hill ; also the use of divergent series that invention of 

 the Devil, as Abel called them - - by Poincare. A particular class 

 of linear differential equations of great importance is the hyper- 

 geometric equation; the results obtained by Gauss, Kummer, 

 Riemann, and Schwarz relating to this equation have had the great- 

 est influence on the development of the general theory. The vast 

 extent and importance of the theory of linear differential equations 

 may be estimated when we recall that within its borders it embraces 

 not only almost all the elementary functions, but also the modular 

 and automorphic functions. 



Too important to pass over in silence is the subject of algebraic 



