368 PROCEEDINGS OF THE AMERICAN ACADEMY. 



#* = ^Qo gx) ^ f ,\ (111; I;;;;;). 



where i/<"' A is continuous in the neighborhood of x = and 



Yi lx=0 - 77— -TT) **> 



K,e„ • 



If the root r K furnishes s elementary divisors : 



(r-r K )\ (r - r K ) e *+\ (r - r K )M-s-l , 



then the constants : 



Ai,K,e K , Ai tK j r i^ K+1 , Ai iK ^. s —i ) e K ^_ s _ 1 (l = 1, 2 . . . n) 



are s linearly independent solutions of the equations (4) when r = r K . 



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

 k = 1, 2, . . . m are linearly independent. 



§7. 



The Homogeneous Linear Differential Equation of the 



nth Order. t 



We shall consider homogeneous linear differential equations which can 

 be written as follows: 



(82) dx^ + U +Pl ) dx'^ + \x +P *)xdaf<-* + -" 



+ (5 + ^)^^°' 



in which ^i 1 , [i„, . . . jt/„ are constants, and p x , /> 2 , . . . p„ are functions 

 of the real independent variable x, continuous in the interval < x 5^ b, 

 and such that their absolute values are integrable up to x — ; in short, 

 these p's are to have the same properties as the functions a u in (1). 



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

 tions by the following substitutions : 



(83) x< % = y-' (» = 0, l,...n-l). 



