Okr — Solutions of Differential Equations by Contour Integrals. 125 



This may be written 



or, by virtue of the given diferential equations, 



fin M^) 

 f^n <p,2W 



Again, by virtue of the differential equations, 



f{t'), i>,,(D') 



A{X)x, 



A{\)x. 



MD'):c 



f{f), 4>,,(I)') 



(49) 



(50) 



(51) 



(the operators in the second, and other,i columns affecting the functions in the 

 first which they multiply, when the determinant is expanded). This is seen 

 by multiplying each equation by the first minor (of A(D')) of its coefficient in 

 x, and adding. 



Thus (49) may be written in the form 



!A(//) - A(A)h' - \<P,,{D') - %,mf{t') 

 - 1%,(D') - <P,^\)]f,{tf), (52) 



where each O is the first minor of the corresponding ^ in A. (I introduce 

 the minors to obtain a form suitable when there are more than two unknowns ; 

 for two unknowns <E>jj = <^22, etc.) And this is true for all values of A. 



But a word of warning is necessary before replacing the determinant in 

 the first term of (47) by the quotient of (52) by D' - A. In the transforma- 

 tions in previous §§ the use of the symbols D, D', has been only to simplify 

 notation ; each transformation lias been purely algebraic. But this is not the 

 case here ; in obtaining (52) we have used the given differential equations. 

 We are not, therefore, necessarily entitled to replace the one quotient now in 

 question by the other. If u, v, are functions of t', we cannot, in fact, from 

 u = V infer u/{D' - A) = v/{l/ - A), unless u, i\ are identical polynomials in 

 D' and A, and each divisible by D' - A ; the two members of the latter equation 

 may differ by a constant multiple of e'^t'. 



Here, however, the two quotients (in the ordinary sense) with which we 

 are concerned are polynomials in A, and, therefore, it cannot be the case that, 

 for all values of A, their difference should be of the form Ce^i'. Hence they 

 must be identical. 



i.e.. when there are more than two unknowns. 



