170 PROFESSOR CHRYSTAL ON THE EQUIVALENCE OF 



factor, equal to the determinant of any system to which it is equivalent ; or, in our 



notation, 



|(11)(22) (nn) \ =[1.1] [22] [nn] . . . (19). 



Every diagonal system may be solved by meant; of a seines of linear differential 

 equations with constant coefficients each involving only a single dependent variable. 

 For we have merely to solve the last equation of (12) to get the complete value of x n ; 

 then, x n being known, the second-last will give the complete value of x tl _ x ; and so on. It 

 will be observed that all the arbitrary constants in the expression for x n (o) n in number, 

 where co n is the degree in D of the coefficient [nn] ) are introduced at once. In finding 

 x n _i we introduce <o a _ x fresh arbitrary constants, where w w _x is the degree in D of \_n — 1, 

 n,— I]. These w n _ x arbitrary constants are the arbitrary constants which occur in the 

 expression for x n _ x , but not in the expression for x n . In addition to these, x n . x may 

 contain all or some of the arbitrary constants already introduced into the expression 

 for a ■ ,. 



Next, we find x n _ 2 by means of a differential equation of degree, <u n _ 2 , x n _. 2 will 

 therefore contain o) n _ 2 uew arbitrary constants, together with all or some of those intro- 

 duced into x n _ x and x„. And so on. 



The whole number of arbitrary constants introduced, none of them superfluous, in 

 the complete solution of the system is o) l + oi. 2 + . . . . +<o a , that is the degree in D of 



[11] [22] [nn]. Hence from (19) we have a rigorous proof of the general 



theorem referred to in the beginning of this communication, viz. : — 



The order of any determinate system of ordinary linear differential equations with 

 constant coefficients is equal to the degree in D of its characteristic determinant. It is 

 obvious that the equivcdence of a diagonal system is not affected by adding to any 

 equation U,. = of the system any linear combination L / ._ 1 U,._i+ . . +L, l U, l = (ivhere 

 L,._!, &c, core integral functions of D with constant coefficients) of all the equations that 

 follow it. For the diagonal coefficient of the resulting equation U,. + L,. _ 1 U,._ 1 + . . + 

 L;|U„ = remains unaltered, and it alone appears in [11] . . . . . [nn], the determinant 

 of the system. Advantage of this may be taken to simplify calculations in the practical 

 solution of a diagonal system. Into the question as to the maximum of simplification 

 thus attainable I shall not enter ; but it appears from the above remark that the 

 coefficients of a diagonal system, other than the diagonal coefficients, are not uniquely 

 determined when the diagonal order of the dependent variables is assigned. 



It is easy, however, to show that the diagonal coefficient of any variable is deter- 

 mined when the aggregate of the variables that follow it in the diagonal order is given. 



For let x,. be the variable in question, x,._ x , x,._ 2 , , x n those that follow it in any 



given order, and let the corresponding diagonal coefficients be [rr], [r— 1, r— l], , 



[n, n] : and let the corresponding coefficient in the case where x,. again stands first, but 

 .'",_i, x,._ 2 , . . . , x n are arranged in any other order be [rr]', [r— 1, r—lj, .... [nnj. 

 Since the last r equations form by themselves in the two cases a pair of equivalent 

 determinate systems for x,. t x,._ h , x„, we must have 



