22 
RADELFINGER. 
easier understanding, a brief summary of some of the earlier 
researches will first be given. 
Old methods .—During the eighteenth century differential 
equations were usually investigated in connection with 
physical or astronomical researches. These equations rep¬ 
resented conditions satisfied by variables occurring in the 
problem in process of solution. It was soon found, however, 
that purely algebraic integrals could not be obtained for 
them, though the non-existence of such integrals was not 
proven. The method of solving then resorted to was as 
follows: 
For the dependent variable was substituted a series with 
undetermined coefficients. These coefficients were then de¬ 
termined by means of the resulting equations. When the 
result was ambiguous, that solution w r as chosen which ful¬ 
filled the other conditions of the problem. The convergence 
of the series representing the solution was not investigated, 
but the validity of the process was tested by instituting a 
comparison with observations. 
Modern methods .—The methods introduced into analysis 
by Abel and Cauchy early in the century, and used with such 
brilliant success in the study of algebraic integrals, were soon 
applied to differential equations. These methods completely 
revolutionized the theory of these equations. 
That but few differential equations are satisfied by known 
functions is a truth now rigidly established. Consequently 
they are now studied in the abstract as defining new func¬ 
tions, the properties of which are determined by the applica¬ 
tion of the comprehensive ideas of the general function 
theory. The theory of these new functions forms the most 
extensive and important branch of modern mathematics. 
In the theory of linear differential equations all investi¬ 
gations are based on the assumption of the existence of an 
integral. That this assumption is true Cauchy was the first 
to prove. His theorem applies to a more general class of dif¬ 
ferential equations, but its adaptation to linear ones is very 
