344 PROCEEDINGS OF THE AMERICAN ACADEMY. 



is called by Weierstrass an elementary divisor of the determinant A (r).* 

 It will be convenient to employ a different notation from that used in 

 the definition of an elementary divisor. An elementary divisor of A (r) 

 will be written : 



(r - r k Y'\ 



and it is to be noticed that several r's with different subscripts, may be 

 equal, as will be the case when a multiple root furnishes several elemen- 

 tary divisors. We shall always have : 



k=m k=m 



ifc=l k-1 



(7) A(r)=E[( r *- r )'* 2* = * 



It can be shown (cf. the next foot-note) that a necessary and sufficient 

 condition that a pair of systems of differential equations : 



, j= n , j=n 



dyi ■__ y \hj ■ t ^i = y^i z . 



dx iH. x dx ^i x ' 



(i — 1 } 2, ... n) 



can be transformed the one into the other by means of a transformation : 



y, = 2^w *j (* = 1, 2, . . . »), 



j=i 



in which the A's are constants whose determinant is not zero, is that the 

 characteristic determinants of the two systems have the same elementary 

 divisors. This theorem enables us to simplify the solution of the system 

 (3) ; for we can write down a second system of differential equations 

 having the same elementary divisors as (3), as follows: 



dz kil _l r k (k = l,2,...m 



W dx -x**-'- l + ^c Zi ' 1 \l=l,2,...e k 



where z k0 = 0. 



* Cf. Muth : Theorie und Anwendung der Elementartheiler, p. 2. 



