( 356 ) 



the second member of the former equation becomes the second 

 member of the equation (1). 

 Let ns now examine 



Vn = j e ' "^ ,c\lx (4) 



b 

 Here too we can find a differential equation satisfied by tliis integral. 

 By differentiating- we find 



1 dv r —^-~~ 1 



2b db J ^2 ^ ^ 



b 



1 d'V„ 1 dV» _ 

 ~ 21^ ~db^ '^ 2^' ~db~ ~ 



b 



wliilst the integration of the identity 



b-\ 62 



x"d \e ^ Jz=z —e •''' x"d.v-\-b''e "" x'^-^dx 



between tlie limits h and oo furnishes 



00 b- (ti b^ 00 b* 



b b b 



So we find for K„ the differential equation 

 d'Vn 2n + lfZF„ 



£^6^ b db 



If we now write the equations {a) and {h) 



dUn 

 db 

 and 



— 4F„zr:(n + l)6»-ie-2^'. ... (5) 

 s [a) and (i^) 



- 2b Un-l (0) 



dVn 



— ^ =. - 26 F„_i - i" >-'-!' (7) 



db 



it is easy to find out of (6) and (2) 



b ^ 



and likewise out of (7) and (5) 



" - 2 ./6 

 Out of the last two equations we deduce the recurrent relation 



b dUn—\ 

 2 db ^ 



b dV„_i />" 



2 db ^ " ' ^ 2 



