BOUNDAEY PROBLEMS AND DEVELOPMENTS. 99 



Y'{x)> = B{x) Y{x) : 

 W a Y(a)- + W b YQ>)> =0. 



(74) „ 7 T//1 , 



In precisely similar manner we have from the adjoint system 



6 



•Z {l \x) = -X ? J-Z (/) (0i?(/)//(.r,/)^, 



a 



or, substituting from the relation 



G(t, x) = - H(x, t). 



b 



(75) -Z (l \t) = h f-Z«\x) R(x) G(x, t) dx. 



a 



Consider, now, the integral 



6 b 



J = J j >Z {l \x) R(x) G(x, t) R(t) Y {k) (t)-dx dt. 



a a 



In view of relation (73) we have 



b 



\ k J = J-Z w (x) R(x) Y {k \x)-dx, 



a 



while it is seen from (75) that 



b 



\ t J= J.Z {l) (t)R(t) Y {k) (t)-dt 



a 



By subtraction, then 



(X,- \) J = 0, 



and it follows that 3 — provided k^p I. But then \J = 0, i.e. 



(76) 







j -Z (l \x) R{x) Y {k \x)-dx = 0, for k =Jz I. 



13 Since X = is not a characteristic value of system (46) system (74) is 

 incompatible and the Green's function exists. 



