Orh — Solutions of Differential Equations hy Contour Integrals. 121 



whatever, provided these have no poles. For these substitutions would 

 give 



2Tri(cl>u{D)x+fr.(Dj)y = 



dXe^* 



^2{X), ^.2(X) 



(29) 



which is zero. 



Again, (25) gives the proper initial values. It gives, as the initial value 

 of ^TtiDfx, a contour integral which differs from that in (25) in having e'^t 

 replaced by unity and the constituents of the first column by the determinant 

 replaced respectively' by 





i\<Pn{I)) - <Pn(\)}^V + \f,,(D) - ,^,,(A))2/]/(Z) - A) 



IH^iiD) - <^.i(A)}.r+ [<^,,{h) - ,t>^..iX)]y-]l{D - A) 



1 = 



(; = o 



(30) 



(31) 



Consider, firstly, the terms in this determinant which involve the initial 

 values of the first unknown, a-, and its derivatives, viz. :- — 



<l>u{ D) - 0n(A-) 

 D-X "" 



t = o 



>n(D)^MAJ J 



. D-X " '^]t = i>' 



<l>n(X) 



(32) 



For the purpose of the contour integration we may subtract from the 

 elements of the first column 



,,.rnp - A^ 

 ^ii(Aii ^;:^ — r a- 



<p2i(X) 



jy - A^' 



D-X 



t = a 



respectively, as this is equivalent to subtracting from the determinant 



A 



(A)[ 



[ V - X" 

 D-X' 



i = 



(33) 

 (34) 



(35) 



and this may be done without altering the integral. 



After these subtractions the elements of the column become 



D-X 



' XP<P2,(D) - Z)^./.3i(A) 

 D-X ' 



t = o 



t = o 



(36) 



(37) 



If p lies between zero and m - 1 inclusiA'e, there is no term in either of 

 higher order than m - 1 in A ; and there is a term of this order in at least 



