868 PROCEEDINGS OF THE AMERICAN ACADEMY. 



y:'' = x^^(\ogxy^-'^:^\ r- 



= 1,2,... n\ 

 — 12 e J' 



1 K, A 



where i/.*' z's continuous in the neighborhood of x = and 



, ^,t^ 



If the root refurnishes s elementary divisors : 



(r-r,)% {r-r,y<+\ (r_r,/<+s-l, 



i^en </ie constants : 



■^j,(c,e^, ^i,K+l,c^_j_i, Ji,K+s-l, e^^.5_l (l = 1, 2 . . . n) 



a?'e s linearly independent solutions of the equations (4) when r = r^. 



The n solutionis of the differential equations (1) that we obtain when 

 K =: l, 2, . . . m are linearly independent. 



§7. 



The Homogeneous Linear Differential Equation of the 



nth Order. 



We shall consider homogeneous linear differential equations which can 

 be written as follows : 



in which ^wj, //j, . . . //„ are constants, and jOi, p^, .../>„ are functions 

 of the real independent variable x, continuous in the interval < x ^ i, 

 and such that their absolute values are integrable up to cc = ; in short, 

 these j9's are to have the same properties as the functions a^jin (1). 



This equation can be replaced by a system of linear differential equa- 

 tions by the following substitutions : 



(83) x^ ^ = 2/„_. {i = 0, 1, ... w - 1). 



