108 PROFESSOR CHKYSTAL ON THE EQUIVALENCE OF 



This second form of the equivalence theorem enables us to test the equivalence of the 

 systems directly, without calculating the system of multipliers. 



Reduction of any Determinate System of Linear Equations with Constant Coefficients 



to an equivalent Diagonal System. 



By a diagonal system is meant a system of the form 



[11>; 1 + [12> 2 + [13>3+ +OK+T 1 = 



[22> 2 +[23]r 3 + +[2n]x n +T 2 = 



[33>; 3 + +[3n]x n +T s = 



H .... 



[nn]x n +T,, = . . . (12), 



where the first equation may contain all the dependent variables ; the second does 

 not contain x x ; the third does not contain x x and x 2 ; and so on, the last containing 

 only one dependent variable, say x n . A diagonal system is characterised by the order of 

 the variables in the diagonal, and there are as many diagonal systems as there are linear 

 permutations of x v . . . ., x n . We shall see presently that the coefficients [11], [22], 

 . . . \nn\ are determined to a constant factor when the " diagonal order " of the 

 variable is given. We shall speak of them as the diagonal coefficients. We shall now 

 shew that 



Every determinate system of linear equations with constant coefficients can be 

 reduced to an equivalent diagonal system in which the dependent variables have any 

 assigned diagonal order. 



In the first place, we prove that 



From any two equations 



U 1 =(ll)a; 1 +(12>: 2 + .... +(lrc>i„+S 1 = . . . (13) 

 U 2 =(21> 1 +(22>c 2 + .... +(2?iK+S 2 = . . . (14) 



we can always deduce an equivalent pair, one of ivhich does not contain any assigned 

 dependent variable, say x v 



For the equations 



LU! + MUj = (15) 



L'l^ + M'U^O (16) 



where L, M, L', M' are any integral functions of D with constant coefficients, will be 

 equivalent to (13) and (14), provided the modulus of (15) and (16) with respect to (13) 

 and (14) be constant — that is, provided 



LM'-L'M = const (17). 



Now, if g be the G.C.M. with respect to D of (11) and (21) ; so that 



(ll)= r/ (ll)' , (21W21)', 



