DUNKEL. — LINEAR DIFFERENTIAL EQUATIONS. 



345 



The characteristic determinant of (8) is 



(9) A (r) 



Ti — r 



1 rj — r 



1 rj — r 



1 Vo — r 



1 r„ — r 



r — r 



1 r.„ — r 



1 r.„ —r 



and it will be easily seen that it has the elementary divisors (r — ?'i)^S 



(r 





(r — r„,)*'". Then, by the theorem above referred to, 



there exists a set of n^ constants -4, ^ ^, whose determinant is not zero, 

 such that : 



(10) 



y> 



k=l 1=1 



i, k, I ■^k, I 



{i = 1, 2, . . . n). 



The system of differential equations (8) we may speak of as the 

 canonical system ; and now it is easily seen that : 



The canonical system of equations admits e^ solutions, corresponding to 



the elementary divisor (r — /■«)*", of the following simple form : 



Zk.i = 



(11) z^^i 



(T^A)-! ^'' (^«g ^> 



i-\ 



I < X 



\<l<e^ 



(X = 1, 2, . . . e^) 



and the n solutions obtained by giving k the values 1,2, ... m are linearly 

 independent. • 



• . — _ 



* Cf. Weierstrass, Werke, Vol. II. pp. 75, 76. The case considered by Weier- 

 strass is very easily reduced to the one we arc considering by the change of inde- 

 pendent variable t = log x. This reduction of Weierstrass is also given in Muth's 

 Eleinentartheiler, pp. 195, 198. On page 198 are a number of references to the 

 .use of the theory of elementary divisors in the study of differential equations. 



