118 BIRKHOFF AND LANGER. 



and it is necessary to consider further only the case x =fc b. We have 



6 



X* 7 k,m,p,q-l ( X ) 



r.^ r - w, - B - i ° s mP + 





b 

 (f *[*&>*&)+*&>-*&% yi (t )m dt) + 



x 



b 



-ll Y „XJ r t (x)-r fc («J + B t (a;)-B.(O s ** /,w /,w s / \ 



XW * i * 8 *fc 7p(0 / P (0 [5*p ^f*W 



X 



+ 8 ik ^ kp (t)]dt\ 



Each integrand in the last matrix of this equation is of the type 

 tpi(t, X) and accordingly the matrix is, by lemma 1 of type (e)/X. 

 Integrating by parts the elements of the first matrix on the right we 

 have 



/«■ = I («« | - e MW(W}^)-^) /i(6 _ 0) +fi{x + 0) + 



Therefore, J2 # can also be expressed as a sum of matrices, the first 

 and third of which, namely 



and 



6 



X 



are of the type (e)/X on arc C M „, for each element either vanishes or 

 approaches the limit zero as | X | = a> . The second matrix of the 

 sum is directly 



