BOUNDARY PROBLEMS AND DEVELOPMENTS. 



121 



irregular conditions considered in section VII. In either case, there- 

 fore, A -1 is uniformly bounded on all circles of type C. 



Proceeding, we have 



Y(x)A~ u - 



( l X ) D(X) 



fc=l 



^iriW-iVWiKl+B^-BiWiii 



flji(X) 

 5(X) 



+ 



f e x\r k (x)-r k (b) 8 Z-\+B k (x)-B k (b)5** k [ya-(3-0]g,fc(X) 



X D(X) 



k=l 



Since the last matrix of this expression is of type (e) we have 



or, more briefly, 



(115) F(a!)A-»= (,xl r *«- r *« M *« * + *i<*>- B .< 6 > 5 « Tt7 (X)), 



where | r t7 (X) | < Jfcf for | X | > A r . ' 



Inasmuch as D(\) was seen to be of the form 



D(\) = W M + 



(f2 



for X on arc C M „, we have for any X on this arc 



, v ( wtf + n \ (wtf\ 



{Til) ~ \w^V^, + V = Vir^V + ( ^ 2) ' 



or, denoting by M " the matrix whose determinant is W {fi "\ 



(116) 



(m») 



(ry) = «<"" + fo). 



17 Cf. Wilder, loc. cit. p. 422. 



