24 
RADELFINGER. 
multiplied by an arbitrary constant, and subject to the con¬ 
dition that between them no relation of the form 
<h Vi 0) + Oj 2/2 0) • • • ..0) = 0 (C) 
subsists, when k£n. 
The general integral is then written in the form : 
y (x) = a x y x ( x ) + a, 2 y % (x) ..... . a n y n (x), (D) 
in which the y’s are particular integrals and the a’s constants. 
In this form of the general integral there are still n arbitrary 
constants as long as no such relation as (C) exists. Systems 
of particular integrals subject to this condition are termed 
fundamental systems. They are of great importance in the 
recent developments of our theory. 
Since Cauchy’s theorem explicitly excludes all singular 
points from the domain within which the existence of an in¬ 
tegral is demonstrated, it gives no information concerning 
the behavior of the integral in the neighborhood of these 
points. For the study of the function within regions con¬ 
taining one or more singular points no general method is 
yet known, and special devices of an inverse nature must be 
resorted to, several of which will be considered later. 
If expansions which converge within a circle described 
about each of the singular points and reaching out to, but 
not including, the nearest one could be obtained for each of 
the fundamental integrals, a complete representation of the 
function defined by equation (A) could be constructed ; for, 
since these circles of convergence would intersect, we would 
have two valid representations of the function within this 
common region, and by making use of this circumstance and 
equating the independent expressions of the function the 
arbitrary constants could be determined by making the in¬ 
dependent variable describe closed paths, including the 
singular points, the constants being determined by elim¬ 
ination from the resulting equations. 
