LINEAR DIFFERENTIAL EQUATIONS. 
25 
These convergent expansions have not been obtained ex¬ 
cept in special cases. These special cases are, however, of 
great importance. The most important one discovered by 
Fuchs is as follows: 
Fuchs’s investigations,* based on Weierstrass’s researches 
on the singularities of functions, were published about 1866, 
which date marks the beginning of an important epoch in 
our subject. Fuchs began by assuming the existence of a 
linear differential equation such that its fundamental inte¬ 
grals could be represented in the interior of circles described 
about the singular points as centers, each circle containing 
but the one singular point, by a series of the form: 
(x — x s )~ m K + a, (x — x 3 ) + a 2 (x — x B ) 2 .], (E) 
in which x s denotes any singular point of the equation and m 
a finite quantity named the strength of the singular point. 
On this assumption he determined the necessary and suffi¬ 
cient conditions that must be fulfilled by the coefficients 
(the p’s) of the general linear equation (A) in order that it 
admit a fundamental system of the assumed form, termed by 
him regular integrals. The general form of the equation 
deduced by him (admitting only regular integrals and having 
the singular points x 1} x 2 , . x s ) is: 
n n — 1 
dy F 1 ( x ) dy 
dx n (x — xf) (x — x 2 ) . . . (x — x s ) dx n ~ 1 
+ 
F n (x) 
(X — xf) n (x — x 2 ) n . . . (x 
y — 0. 
(F) 
The F’s in this equation are rational algebraic functions of x. 
They are not entirely arbitrary, but are subject to the condi¬ 
tion that the point at infinity be an ordinary one or a singular 
point of only finite strength. Fuchs also proved the con¬ 
vergence of the series representing the fundamental integrals 
of equation (F) within the assumed domains. 
* Crelle’s Journal, vols. 66 and 68. 
