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VII. 



SOLUTIONS OF SYSTEMS OF OKDINAHY LINEAR DIFFERENTIAL 

 EQUATIONS BY CONTOUR INTEGRALS. 



By PROFESSOR W. M'F. ORR, D.Sc, F.R.S. 



[Read .ii-iai. 23. Published JuLV 2S, 1923.] 



In a paper on E.xtensions of Fourier's Theorem,' etc., I have virtually used, 

 incidentally, the solutions given below. They may be said to differ only in 

 notation from those given by Routh.^ 



The usual forms of solution may, of course, be obtained by replacing the 

 contour integrals by sums of residues. 



It is not easy to show rigorously what is the most general solution of a 

 system of simultaneous differential equations, (linear, and with constant 

 coefficients). The treatment of this point in text-books is unsatisfactory. 

 Even Routh's highly instructive, and otherwise full, discussion does not meet 

 this issue. The question has been satisfactorily discussed by Chrystal.^ 



I consider it an interesting feature of this paper, and perhaps the main 

 justification for publishing it, that it not merely gives a solution, and this in 

 a compact and comprehensive form, but proves that it is the solution. It 

 would, however, prove a meagre substitute for Routh's and Chrystal's dis- 

 cussions. 



" And, although I prove, rigorously, and, I think, simply, that I obtain 

 the most general solution, I have not succeeded, except in one case, in 

 proving what is the number of independent constants wliich it contains, that 

 is, without recourse to Chrystal's argument, or a very similar one. The 

 exceptional case is that in which the order of the characteristic determi- 

 nant, A, is equal to the sum of the orders of the system in each of the 

 unknowns separately, understanding, by the order in any particular unknown, 

 that of the highest derivative of that unknown which occurs anywhere in 

 the system. When this equality holds, I shall, for convenience, speak of A 

 as " normal " ; and, when A is of order lower than the sum of the orders in 

 the unknowns separately, I shall describe it as " abnormal." 



1 " Extensions of Fourier's and the Bessel-Fourier Theorem " : Second Paper, Articles 

 5-8, P.R.I.A., 1911. 



2 "Dynamics of a System of Rigid Bodies." 



5 "Equivalence of Systems of Ordinary Linear Differential Equations," Trans. Rpy. 

 Soc. Edin., xxxviii, pp. 163-178. 



B.I.A. PKOC, VOL. XXXVI, SECT. A. [18] 



