BOUNDARY PROBLEMS AND DEVELOPMENTS. 105 



Now it was seen (page 87) that at a characteristic value D(\) 

 vanishes. Let us assume further that the characteristic values of 

 system (46) are all simple so that Z)(X) vanishes at each of these values 

 to only the first order. Due to the relation 



(87) | Da | = D 



n-l 



which is familiar from the theory of determinants, 15 it follows, then, 

 that Dij(\k) cannot vanish for all i and j, since in the alternative case 

 the left-hand side of (87) would vanish to the order n, while the right- 

 hand side vanishes only to the order (n — 1). Hence (Z),-i(XjO) 4 1 0, 

 and A _1 (X) has a simple pole at each of the characteristic values. 

 Moreover, system (46) is simply compatible at the characteristic 

 values as was assumed above. It is easily verified (i), that the poles 

 of A -1 (\) actually persist in G{x, t, X), and (ii), that G(x, t, X) has no 

 other singularities. 



Denoting by G {k) (x, t) the residue of G(x, t, X) at X = \u we obtain 

 from (86) by familiar methods 



G k (x t) = Y w (x) (DiiM) \W a Y w (a) - W b Y (k \b)}Z\t) 



d 



*~h 



which, inasmuch as — D(\) 



c/X 



*=h 



=1= 0, is of the form 



(88) G' w (.r, t) -- Y ik \x) C {k) Z {k \t). 



From this it follows that for fixed /, G (x, t) is a solution of the system 



(89) \Y'(x)= {R(x) \ k + B(x)} Y(x) 

 (W a Y(a) + W b Y(b) = 0. 



But if Y(x) is a particular solution of the differential equation of this 

 system every solution is of the form 



Y(x) = Y(x) C. 



Substituting in the boundary conditions as on page 64 and denoting 

 by (dij) the A which results from Y = Y we have 



15 Cf. Bocher, loc. cit,, p. 33. 



