BOUNDARY PROBLEMS AND DEVELOPMENTS. 87 



In the course of the deduction of forms (45) it was assumed through- 

 out that n ^ 2. Direct integration of the equation, however, shows 

 that in the case n = I there exist solutions which are of this form over 

 the entire plane. Accordingly formulas (45) may be used in every 

 case. The explicit forms 



yW= ^r,(,) +B ,. W j i , 7 + ^>j), 



with (pij, \pij bounded, hold if R(x), B(x) are differentiable. 



Section VII. 

 The characteristic values of the system 



Y'(xj. = {R(x)\+B(x)}Y(x). 



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



It was shown on page 65 that a necessary and sufficient condition 

 that the vector system 



K *° } W a Y(a) ■ + W b Y(b) ■ = (| W a | * 0, | W b | * 0), 



has a solution is that 



(47) | W a Y(a) + W b Y(b) \ = 0, 



Y(x) being any matrix solution of equation (26). Let us make the 

 specific form of this condition in any sector of the X plane apparent by 

 substituting a Y(x) which is analytic in X for | X | > N, and which has 

 within the sector the form 



(48) Y(x) = (e*V x)+B > (x) [S i} ]) , 



as determined in the preceding sections. Choosing the a of formula 

 (20) as a = a it follows that Y(a) = ([5 l7 ]). 



Substituting from (48) in the determinant on the left of equation 

 (47), (call it D{\)) we have 



