BOUNDARY PROBLEMS AND DEVELOPMENTS. 93 



from (61) the product II e l xr V 6)+i W 5 u- we have 



k=i 



** 



(62) b(X)- fi e |Xr ' <W+S ' (W l iS 5(X), 



where 



+ 5 ^ e xr J , 2 (6) + s„ ! (6)|| i 

 The determinant Z)(X) is seen, therefore, to have monomial elements 



fh fh 



in each column except the v x and the v 2 , the elements in these 

 columns being binomials. It is clear, therefore, that D(\) can be 

 expressed as the sum of four determinants each containing only 

 monomial elements, i.e. 



(63) S(X) = | [«{?] s£ + WSS] a£? I + 1 [«!?] {*£? - «„, } 



** 



* ** 



* 



I [Vre ](5 C c — 5 C ^ — <5 C „J 



** 



Let us again denote by a> and o-„ T the sectors abutting on the ray 

 R{\T Vi (b)} = in which fljxr,,^)} < and > respectively. It is 

 necessary to consider the case in which arg I\ (6) = arg r„ (b) and 

 that in which arg T„ (b) = arg { — r„ (b) } . 



Sub-case A. arg T„ (b) = arg T Vi> (b). 



In this case the quantities i?{Xr^(ft)} and i?{Xr„ (b)} are of the same 

 sign throughout the sector S, and the relations 



(64) 





