BOUNDARY PROBLEMS AND DEVELOPMENTS. 



107 



Y {k) (x)- = - {X - X,} J G(x, t, X) R(t) Y {k \t)-dt. 



Inasmuch as 



this yields 



lim {X-X,} G{x,t,\) = G {k) (x,t), 

 x=x fc 



o 



Y {k) (x)- = - J G {k \x,t)R(t) Y {k) {t)-dt, 



and substituting from (92) we see that 



6 



Y {k) (x)- = - — Y m (x)--Z (k) (t)R(t) Y {k \t)-dt, 



*J <n 



But in view of the relation (85), 

 b 



Y {k \x)-f-Z {k \t) R(t) Y {k) (t)-dt = 



a 



f I 



n 



fzz ( h k \t)y h (t) 



(k). 



} Y {k) (x) 



It follows that 



1 - - c 



(k) 



6 



f 2 *P 



(k). 



(t) y h (t) y nt) dt, 



or upon associating with F ( (.r)- the proper -Z { \x) (see page 101) 



thatc (fe) = -1. 



Hence 



(93) G (k \x,t)= - - Y {k) (x)--Z {k) (t). 



n 



This result enables us to deduce an expression for the sum of m 

 terms of the series development of an arbitrary vector F(x) • . Thus 

 we have from (77) and (78), (-Z^and Y l • being associated in the 

 manner stated) 



