90 BIRKHOFF AND LANGER. 



But it is readily seen that 

 (54) 8& = «<£? - 8 e „ 5$ = 8$ + S ev . 



Hence if S is any sector which includes the ray R[\T v (b)} = and 

 overlaps a part of each of the sectors a ^ and o VT we have, for a X in S. 

 the following expressions 



D(\) = | [««] 8Z + [w { »] 8 { B e^'M+BcMl for X within <x M „ 



* ** 



D(\) = | [««] tt? + MS M? + U^W+B,^ for X on the 



ray R{\Y v (b)\ = 0, 

 * ** 



DW = I [«%] la<S? - U + [w<g] (atf +*»} ^«+«*>|, farX 



within & VTJ 



It is readily seen from this that D(\) is given for any X in S by the 

 formula 



(55) D(\) = | &?] $ + [*g2][$ + t m \**M+*M\. 



n ** 



Factoring from this determinant the product II r Ur*(6) + B*(6)| S( t £) 

 we have 



(56) D(\) = ri e M r *»> +**<*> ) 8 h? Z)(X), 



fc-i 

 where 



5(X) = I [«{?] $ + k^]{5^ > + 5 C „} ^r„(6)+B,(» |. 



Now D(X) is seen to have elements consisting of a single term in every 

 column but the v , the elements of that column being binomial. Con- 

 sequently the expression of the determinant as the sum of two others 

 is possible, i.e. 



D(K = | [«#>] 8[f + [«£>] S& | + | M?)lst - S c „} + 



** 



or, in view of relations (53) and (54), 



(57) D(\) = [JFH -f [JV T ] e r„x(w+B,(» 



