116 BIRKHOFF AND LANGER. 



Accordingly it is seen that Jy can be considered as a sum of ma- 

 trices of which the first three are discussed as follows. 

 For the first matrix we have directly 



\ (8*Mx-0))= I (4)^-0).. 



A A 



The second matrix 



- I fa ******* M*)) 



A 



is one each of whose elements is seen either to vanish or to be of type 

 ^> 2 (X)/X on arc C M „. Hence the entire matrix may be represented 

 by M/\. 



The third matrix 



X 



- ^&- JVl^-W | ifi(t) e B i (l) ~ B i (t) }dt) 



a 



is likewise one each of whose elements either vanishes or is, on arc 

 C n V , of type <£>2/a by lemma 3. This matrix is, therefore, also of the 

 type (<P2)/\ and inasmuch as a sum of matrices of type (^2) is again 

 a matrix (</? 2 ) we have 



Jv= ^ l0S)>(*-O).+ fo)) 



The integration of J\ • over the arc C„„ yields, therefore, by lemma 2 



Cav V 



)f - = °^ (5Sf ] ) F(x • (e), 



where w M „ is the angle subtended by C M „ at X = 0. 



A similar expression results from integration over each arc similar 

 in character to C„„. Hence we have 



* 



sff(«) • = Z =c (ajr) ) f(x - o) • + (e). 



c 2tt 



Let us consider in detail the sum 





(107) L^(8\f). 



C 2ir 



