LINEAR DIFFERENTIAL EQUATIONS. 
BY 
Frank Gustave Radelfinger. 
[Read before the Society May 27, 1899.] 
Introduction .—Felix Klein, the great geometer of Got¬ 
tingen, has said that the central problem of modern mathe¬ 
matics is the development of the properties and interrelations 
of the functions defined by differential equations. It is in 
this large and constantly widening field that most of the 
great mathematicians of the past thirty years have labored 
and achieved their most important successes. All methods, 
old and new, geometrical and analytical, have been employed. 
Every new idea has been eagerly caught up and industriously 
applied. Therefore the task of digesting this mass of results, 
obtained from so many and such diverse points of view, is an 
especially difficult one. 
While a consistent theory of linear differential equations 
has been only partially established, this theory represents 
but a small part of the progress made in the establishment 
of a theory of differential equations in general. The meth¬ 
ods employed in the study of the algebraic functions and 
their integrals are more easily applied to linear differential 
equations than to any other class; consequently this theory 
is the most complete. The theory of a special class of them 
(Fuchs’s class) already rivals in completeness that of the 
Abelian integrals. 
This paper will be limited to reviewing only some of the 
most recent advances in this interesting subject; but for its 
4—Bull. Phil. Soc., Wash., Vol. 14. (21) 
