102 



BIKKHOFF AND LANGER. 



In a precisely analogous manner it may be found that a development 

 of the type 



■F(x) = Lc k -Z ik) (x), 

 fc-i 



which converges in such manner that it may be integrated term by 

 term after being multiplied by any of the vectors R(x) Y (%)•, must 

 necessarily coincide with the expansion 



00 



k-1 





)y h (x)y^(x)dx 



Z w (x). 



Inasmuch as the matrix G(x, t) is not a vector the methods outlined 

 above do not apply directly to the problem of expanding the Green's 

 function. We shall proceed, therefore, as follows. 



Let Gj(x, t) • denote the vector each of whose columns is the j col- 

 umn of G(x, t) . From the relation (SO) , F(x) being replaced by Gj(x, t) • , 

 it follows that for any value of t, Gj(x, t) • can be formally developed, 

 the series obtained being 



(81) Gj(x,t) 



fc=i 



6 

 d h-1 



r) 7h(x) (I hi (x, dx 



<k) 



Y w (x).. 



In view of equation (75) written in the form 



(z { /\t)) = \ t f 'iz h l) (x)y h (x)g hj (x,t)dx. 



'' h-l 



(81) reduces to 



OO (ft)/,N 



k=l *k 



from which we obtain the relation 

 (82) 



gaix, = 21 r~ 



