126 BIRKHOFF AND LANGER. 



The second matrix of the sum on the right, namely 



A \k.h-l ' 



is seen to be of the type {<p->)/\, whereas the third, i.e. 



1 

 X 



k,h=l 



** 

 ii 







^ 



is also of this type, as is apparent from the fact that by lemma 3, each 

 of the integrals with respect to t which actually occurs in any element 

 of the matrix is of type <p-i, while the remaining factors are bounded. 

 The remaining matrix of importance in the expression for Jf, 

 namelv 



J 4 (x)- = — 



k,h=i 



again depends for its character upon the value of x, being of the type 

 (<?2)/X when and only when x ^z a and x^pb. By reasoning now 

 familiar the results 



J 4 (a) • = ^ ( £ 5« ru(X) 4a S; A /fc(6 - 0) + *> 2 ) 



= ^ (4) S2- 1 IF 6 (5* ; ) F(b - 0) • + ^ W, 

 and 



J 4 (6) • = — [LCi ra(\) «$ *I» A (6 - 0) + ft j 



X 



U-,A=1 



~Y L (*J) tt- 1 FP» (S^) F(6 - 0) • + i (ft), 



