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II. — A Fundamental Theorem regarding the Equivalence of Systems of Ordinary 

 Linear Differential Equations, and its Application to the Determination of the 

 Order and the Systematic Solution of a Determinate System of such Equations. 

 By George Chrystal, M.A., LL.D., Professor of Mathematics in the University 



of Edinburgh. 



(Read 18th February 1895.) 



Systems of ordinary linear differential equations are of great importance, both from 

 a practical and from a theoretical point of view. They figure largely in dynamical 

 problems ; and Jacobi has shown that the general problem of determining the order of 

 any system of ordinary differential equations whatever can be reduced to the problem of 

 determining the order of a linear system with constant coefficients. Nevertheless, the 

 present state of the theory of such a system still leaves something to be desired. It is 

 true that a logical and systematic process for the solution was given by Cauchy. This 

 consists in first replacing the system by another in which only first differential co- 

 efficients occur, by introducing as auxiliary variables the successive differential coefficients 

 of the various dependent variables up to the highest but one, and then reducing this 

 system to the "normal form" by calculating the differential coefficients as linear 

 functions of the dependent variables. It happens, however, when we attempt to do 

 this, that we are led to a system consisting partly of differential equations of the form 



^=f r (, v ; x,)+g t (t) r = l, 2, ,s. . . . (1), 



where f denotes a linear function of x v . . . ., x s , partly of a number of non- differential 

 equations connecting the remainder of the variables with x v . . . ., as g . The order of the 

 system — that is to say, the number of independent arbitrary constants required for 

 its complete solution — is the number of differential equations in the normal form ; but 

 no rule is readily deducible from the method for determining beforehand how many of 

 the equations in the normal system will be differential equations, so that we cannot 

 predict the order of the system without actually going through the labour of reduction. 

 Moreover, the normal form is in practice often not the most convenient for the purposes 

 of solution. 



Another method, the one which is probably more familiar to English mathematicians, 

 consists in using what may be called the " characteristic equation " of the system. For 

 example, let x, y, z be the dependent variables and t the independent variable, and let 

 the system be 



f 1 (D)x+y 1 (J)) 1/ +h 1 (B)z = ) 



f 2 (D)x+y 2 (T))y + k 2 (D)z = 0[ .... (1) 



f 3 (B)x+ ffs (D)y+h i (J))z=0) 



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



