124 



BIRKHOFF AND LANGER. 



1 * -1 ** 



Jz(a) ■ = — (*&*) n M W a (8\f) F(a + 0) • + fa) } 



A 

 -1 



** —i ** 



Mb) • = — I <WT) " ^.(*T) ^(a + 0) • + fe> 2 ) } . 



A 



It follows from this that 



1 / j\ 



= Z — . J 00 — when - r ^ a > x ^ & - 



:(3) 



i 2iri r 



-1 



O*)- \ 



1 I * ** r/\ 



= L —J {-tif)Q M Tr a $?V(a+(». + (**)}- 



when .r = a 



1 I ** -1 ** TV 



= I f-. J i - &?) &* w a (ajr>) F(« + o) • + fa) } =* 



c 2?ri <V X 



when .r = b, 



or, upon defining the matrices Ks and iv*3 by the equations 



K 3 = £ 



1 'i/i ^^ 



(117) 



_'7T 



( >r ** -1 ** ) 



I K 3 = z - — $f ) " w ° tif) . 



[ c ( 2tt ) 



H3) 







when .r 4 1 <?, ■*" 4 1 6. 



(118) lim O*)- = A r 3 F(« + 0) • when x = a. 



= f 3 .F(a+0)- when a; = 6. 



It should be observed that the values of Kz and Kz depend only 

 upon the differential system whose characteristic functions form the 

 terms of the expansion in question and are in particular entirely inde- 

 pendent of the vector F(j) • 



iv. sL%)- 



Proceeding precisely as in the case of S„ (x) • we have upon defining 

 Ji ■ by the relation 



