Okk — Solutions of Differential Equations by Contour Integrals. 127 



Transferring the third term to the right-hand side of the equation, we obtain, 

 as the solution for x, 



2irixt, = Q 



A(A) 



MX) 



{Kii>) - ^,iW\ 



I)~X 

 D-X 



^22(^)1 y 



</>12(A) 



e-M'f^{t']dt', 

 c-^ffjt') dt', 



<f,,,(D) 



4>-2m 



t = 



(57) 



The second term on the right is thus a "Particular Integral"; in it, if 

 there are more than two unknowns, D occurs in all elements save those of 

 the first column, and the operations indicated by functions of D are to be 

 performed on those elements of the first column which they multiply. It may 

 be well to emphasize that D in this term indicates differentiation with respect 

 to t performed after integrations with respect to t'. 



In the corresponding value of y, the first columns of the two determinants 

 in (57) become the second columns instead. 



The form which first suggested itself Lo me as a solution differed from (57) 

 and its analogues in having I) in the second term replaced by A. This gives 

 the same value for x when A is " normal," but usually not in the contrary case. 



§ 4'2. The solution (57) satisfies the differential equations, and does so if the 

 constants are replaced by any whatever. 



If, in the first determinant in (57), the constants which represent the 

 initial values of the unknowns and their derivatives are replaced by any 

 others whatever, the differential equations are still satisfied. (But it is not 

 the case that in such a form of solution these constants necessarily represent 

 initial values of such unknowns and derivatives when A is " abnormal." (See 

 § (4'4), below.) 



In the first place, if we substitute in the equations (44), (45), for x, y, the 

 values given by the first terms of (57) and its analogues, the left-hand 

 members vanish ; this follows as in § (3-1). 



It remains to be shown that the second terms give " Particular Integrals." 

 When the values of x, y, given by these second terms are substituted in (44), 

 the left-hand member becomes 



{2TviY 



but, from (20), 



'^'^ . A(i^) 

 A(A) '^^ ^ 



cHi-nf{t')dt'; 



A(i)) 



eHt-t')f{t')dt' = AiX) 



6Hi-nf(f)dt' 

 



A(A) 



Z» 



/i(0. 



(58) 



(59) 



