DUNKEL. — LINEAR DIFFERENTIAL EQUATIONS. 



345 



The characteristic determinant of (8) is 



(9) A (r) = 



r x — r 



1 r 1 — r 



1 r x — r 



r 2 — r 

 1 r 2 — r 



1 r„ — r 



r m — r 



1 r,„ — r 



1 r — r 



and it will be easily seen that it has the elementary divisors (r — r-^f 1 , 

 (r — r 2 ) et , ... (r — r, n ) e '". Then, by the theorem above referred to, 

 there exists a set of n 2 constants A lkJ , whose determinant is not zero, 

 such that: 



(10) 



A=l Z=l 



(« = 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 K solutions, corresponding to 

 the elementary divisor (r — r K ) 6K , of the following simple form : 



(11) z K ,i=0 



z K ,i = 



(l-X)\ 



x T « (log x) 



l—\ 



I < A 

 A < I < e K 



(A = 1, 2, . . . e K ) 



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 are considering by the change of inde- 

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

 Elementartheiler, 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. 



