770 



We shall also apply Volterra's inetiiod to (1) and so we shall 

 be led to important conclusions. 



1. We premise the well-known relation'): 







substitute in it: a ^ ,?/==:; i^.x- — a, and replace v by — X, q by 



.V -* a 



n -\- in (;n and m positive integers or zero), so as to get : 

 I„+m+i-). (sJ/.«— «) = 



"+'"+1 



— I I an 



We multiply both members of (4) by —I I and summate 



afterwards from ?2 = to ?z=go. The first member may be reduced 

 by means of the relation ") -. 



,(!)", ,, (iJ 



„=o nf "+ '■" r(v+l) 

 and in the second member summation under the integral sign is 

 allowed on account of uniform convergence of the series arisen. 

 After some reduction we find : 



[2) ^^-'^"-''-' 



r{vi-\-2—x) 



(5) 



= r~Y ƒ(.!—§) 2 j_, {zV/JZr^).{^-a)-U,„ (iz\/^-a)dl «(;.)<l.l 



If we further substitute in (3) c< =: ^, i/=.k\'^.v — (7, and replace 



;v — a 



V by [x — 'J, Q by m — ft {m again positive integer or zero), we get : 



1) Nielsen: Handb. der .Gylinderfunktionen 1904, p. 181 (8) slightly altered. 

 2; Nielsen. 1. c. p. 97 (6). 



