104 



BIRKHOFF AND LANGER. 



or, integrating term by term, 



(84) 



F(z). = EF w Gr) 



k~\ 



o 



-J- 



'(k) 



Z w (jt)C(t)'dt 



Now given any matrix of type D • , and another of type • A • , we have 



J) -A- = (2 did) = (nadi), i.e. 



(85) 



Hence (84) reduces to 



F(x) 



k-l 



D-A- = naD- 



b 



fzzi k) (t)c h (t)dt 



Y w (x)., 



the series on the right converging uniformly to F(x)- Q. E. D. 



In formula (12) the explicit expression for the Green's function 

 G(x, t) for an incompatible system of type (6) is given, Y h and Z h being 

 respectively a solution of the differential matrix equation l 7/ (.r) = 

 A(x) Y(x), and of its adjoint. Now for any X, not a characteristic 

 value, system (46) being incompatible is precisely of type (6). Conse- 

 quently we have from (12) 



(86) G(x, t, X) = G(x, t, X) - Y{x, X)A" 1 (X) { W a G(a, t, X) +_ 



W b G(b,t,\} } 

 for any X not a characteristic value. 



Let us investigate now the functional dependence of G(x, t, X) upon 

 the parameter X, choosing as the solutions Y(x, X) and Z{t, X) a pair 

 which are analytic in X. Recalling that when the determinant of a 



matrix A is denoted by A then A~ 1 = (~r), where An is the cofactor 



of the ij th element of A, the explicit form of A-'(X) is seen to be 



14 This notation holds when n = 1 provided it is agreed that when A is a 

 matrix of one element then ,4 n ~ 1. 



