SYSTEMS OF ORDINARY LINEAR DIFFERENTIAL EQUATIONS. 171 



[rr][r-l,r-l] [nn] = [rr]'[r-l, r-lj [nn]' . , (20) 



by the second equivalence theorem. Again, the last r — 1 equations in the two cases 

 form a pair of equivalent determinate systems for cc,._ l5 x r _ 2 , , x n ; hence 



[r-l,r— 1] [?m] = [r-l, r-1] [nn]' . . (21). 



Now, from (20) and (21) we have [VrJ^JV, r]', which proves our theorem. 



It is obvious, alike from considerations already detailed regarding the successive 

 introduction of the arbitrary constants, and from the possible derivations by means of 

 which we can deduce from a given diagonal system an equivalent one with a different 

 diagonal order of the variables, that the diagonal coefficient for any given variable is 

 of least degree in D when it is first, and of greatest degree when it is last in the 

 diagonal order; and that promotion in the diagonal order' may increase but cannot 

 diminish the degree of the diagonal coefficient. 



The diagonal coefficients in the two extreme cases are of greatest importance ; because 

 the degree of the diagonal coefficient, when the variable is last in the diagonal order, is 

 the whole number of arbitrary constants in the complete expression for the dependent 

 variable in question ; and the degree, when it is first, in the number of arbitrary constants 

 which occur in the expression for that variable and do not occur in the expression for 

 any of the others. We are thus led to investigate rules for calculating the first 

 and last of the diagonal coefficients for any given order of the dependent variables, say 



£,.-■■,& ! &', • • • 



Ki ,...., Kn 



fc» 



f l > • • • • j Kji 



be the systems of multipliers of (12) 'with respect to (18), and of (18) with respect to 

 (12), where, since the systems are equivalent, all the multipliers must be integral 

 functions of D. Then we have, inter alia, 



[11] £'=(11), [11] 9l '=(21), , [11] Kl ' = (nl) . . . (22) 



<11)£ + (21)&+. . .+(nl)& = [ll] .... (23) 



From (22) it follows that [11] must be a common divisor of (11), (21), , (nl) ; 



and from (23) that the G.C.M. of (11), (21), , (nl) must divide [11] exactly. 



Hence [11] must, to a constant factor, be simply the G.C.M. of (11), (21), , (nl), 



!h say. 



Again, the complete system for determining k x , , /c„ is 



(11) Ki + (21)k 2 + . . .+(nlV„ = 



[ . . . . (24). 



(l,n-l) Kl +(2,R-l) K . 2 + . . . +(n,n-l) Kn =0 ' 



(ln) Kl + (2n)K 2 + . . . +(nn)K n = [nn] 



VOL. XXXVIII. PART I. (NO. 2). 



