BOUNDARY PROBLEMS AND DEVELOPMENTS. 85 



n 



i.e. ^ , 



c a — ■" Pih ( >hj , 

 h=\ 



where the quantities p;/, are analytic in X. 



Inasmuch as Y( x) is a non-identicall y zero solution if D =(= there 

 will exist such a solution for which C — provided the determinant 

 | p^ | vanishes. If on the other hand this determinant does not 

 vanish then there exists a solution Y(x) ^ corresponding to every 



Suppose the determinant | p ih \ = . Then there corresponds to the 

 choice C = a solution Y(x) for which the matrix Uix) defined by 

 Y(x) = U(x) S(x) satisfies the relation 



i / » rx/{ 7A (f)- 7r ( f )}di \ 



But we know that for some x, say xo, and for some i, j, say io, jo, 

 u ujo( x o) — M. Then since 





\L et u>%?(xo,t)dt\<KM, 



n,r 



KM 

 we have for the io, jo element, M < — — , 



X 



i.e. m(i-~}<0. 



Inasmuch as M is positive and X can be taken arbitrarily large this 

 involves a contradiction, and proves the hypothesis untenable. 

 Hence | p^ | =}= 0. Accordingly there corresponds to every choice of 

 C 3= 0, and hence to the particular choice C = I, a unique D 4 1 0, and 

 hence a solution Y(x). By equation (44) the U(x) for this Y(x) must 

 satisfy 



U(x) = I - ~ f(x, X) . 

 A 



We infer from this the inequality, 



KM 



