SYSTEMS OF ORDINARY LINEAR DIFFERENTIAL EQUATIONS. 169 



where (11)' and (21)' are integral functions of D, which are prime with respect to D, and 

 we take 



L=(2iy , M=-(iiy, 



then, by a well-known theorem (see my Algebra, vol. i. chap. vi. § 11) we can always 

 determine two integral functions of D, L' and M', such that 



(21)M'+(11)L' = const. 



If L, M, L', M' be thus determined, the coefficient of x 1 in (15) will vanish, and the con- 

 dition (17) will be satisfied. Hence our first preparatory theorem is established. 



We next prove that 



A determinate system of linear equations iviih constant coefficients can always be 



replaced by an equivalent system in which any given variable, say x 1} occurs in only one 



of the equations. 



Let the system be 



^=(11)^+ +(lttK+S 1 = .... (18) 



U n =(nl)x 1 + +(nn)x n +S n = Q . 



If any of the equations already do not contain x x , set them aside and consider those 

 that do contain x 1} say the first r. By our last theorem, we can replace U], = 0, U 2 = 

 by an equivalent pair, U/ = 0, U 2 ' = 0, one at least of which does not contain x x . Set 

 that equation, say U/ = 0, aside along with the others that do not contain x x . If U 2 ' = 

 happens not to contain x 1} set it aside also : if not, take it along with the remaining 

 r - 2 equations. We thus have a system of r—l equations, with which we can deal as 

 before. By continuing this process we shall finally arrive at a single equation which 

 must contain x x , since the original system is determinate. This last equation, conjoined 

 with the n — 1 equations set aside in the above process, constitute a system equivalent 

 to (18), only one equation in which contains x v 



The possibility of reducing any given system (18) to a diagonal system is now obvious. 

 We arrange the dependent variables in any order, say x 1} x 2 , x s , . . . . , x n ; then deduce 

 from (18) an equivalent system only one equation, say the first, of which contains x v 

 The remaining n—\ equations form a determinate system for x. 2 , x s , . . . . , x n . From 

 this last deduce an equivalent system the first equation of which alone contains x 2 ; and 

 so on. We thus arrive finally at a diagonal system, such as (12). 



Properties of a Diagonal System. 



It is immediately obvious that the determinant of a diagonal system reduces to the 

 product of the diagonal coefficients. Hence by the second form of our equivalence 

 theorem it follows that 



The product of the diagonal coefficients of any diagonal system is, to a constant 



