SYSTEMS OF ORDINARY LINEAR DIFFERENTIAL EQUATIONS. 165 



The theorem in question is true ; but the difficulty in demonstrating it lies in 

 the fact that although each of the variables satisfies the characteristic equation, all of 

 them may not be general integrals of that equation. In fact, it may happen that 

 no one of them is a general integral. This may be seen at once by supposing the 

 integral system of (1) to have the form x=f(b, c, t), y = g (c, a, t), z — h (a, b, t), where 

 a, b, c are arbitrary constants. In this case the order of the characteristic equation 

 is 3 ; but the order of the differential equations which determine the separate variables is 

 in each case 2. 



I propose in this communication to give a rigorous proof of the general theorem 

 above referred to, by means of a simple theorem regarding the equivalence of systems of 

 linear differential equations with constant coefficients, and to deduce a systematic 

 method for solving determinate systems of this kind which does not introduce superfluous 

 arbitrary constants, and is not subject to failure in particular cases. 



Necessary and Sufficient Condition that tivo Systems of Linear Equations with 

 Constant Coefficients be equivalent. 



Let the dependent variables be x u . . . ., x n , the independent variable t, and let 

 U r and V r be expressions of the form, 



(r, 1) x x + (r, 2)x 2 +.— +(r, n)x n + S r , 

 [r, 1] ^ + [r, 2].x 2 + . - + [r, n]x n + T, , 



where (r, 1) (r, n), \_r, 1], . . . , [r, n\ are integral functions of D ( = d/dt) 



with constant coefficients, and S,. and T r are functions of t alone. 



Consider any original system of m independent equations (m > n). 



U 1= 0, , U w = (5). 



Let 



^=0, , V TO =0 (6) 



be a system of m independent equations " derived from " (5) — that is to say, possessing 

 the property that every solution of (5) is a solution of (6). 



Since the derived system is linear with constant coefficients as well as the original 

 one, any process of derivation must consist in operating on the equations of the original 

 system with integral powers of D, multiplying the resulting equations thus obtained by 

 constants and adding. Hence we must have 



y 1 =&Ui+ • •■ — +g m u m 



V 2 = J7 1 U 1 + ...... + Vm Ur, (7) 



V m — «r 1 U 1 + +K m V m 



where £ 1} . . . . , £ TO , , k x , K m are integral functions of D with 



constant coefficients. We may speak of these as the midtiplier- system which derives (6) 

 from (5) ; and call the determinant 



