( 355 ) 



Uu 



Tlii.s ditierential equation we also find if we put .v = 2z 6 and v = — — 

 in Bessel's equation : 



dx^ w dx \ A' J 



therefore 



Uu = ^."+1 [A /"+' {21 b) + B r"+i {21 b)]. 



In order to determine the constants properly I notice that the 

 integral £/"„ for /; = is equal to nf and vanishes for 6 = oo ; 

 moreover we find 

 for ^^ = 6«+i/'=+i(2i6) = 0, 



6«-fi r"+i (2i 6) = — (- 0"+' — , 



Ju+1 2b-{-^— 



for ^ = 00 &"+! / "+1 (2i ^) z= — — = e " , 



tlms 



i«+l pi+1 (2i M = -^ e 



= A4-Be — , 

 ^ 2 



and tinallv 



Un = n t"+2 6"+i [/"+! (2/ ^) + t r "+i (2/ 6)] == jr/"+2 bn+i Hn-\-\ (2^ 6) 1) . .(3 



That this value and the ^'alue (i) agree is easy to prove. For 

 according to definition ^) we find : 



jt r "+' (2/ b) — 2 /"+1 {21 b){lqb-\-—\-~(—\ 2 ^ ^ ^ &2., _ 



- (.• Z>)"+i V - ^^" [t|, {s + 1) + t|. (. + n + 2)J , 



from which ensues when we multiply by <"+^ /'"+^ 



;/ 1"+2 />'*+! [/n+1 (2^■ /O + i r«+" (2/ ^>)] = 21^'+^ /V'+l A/ A /"+1 (2i 6) -f 



+ ^^ —, ^ ^^^ -(-!)" ^^"+--^ -77— -TlT/r-V'(H-l) + ^f'(^ + n + 2)]. 



By 



ƒ"+! (2/ b) — {I 6)"+i 2 , 



1) Nielsen, Handbuch der Cylinderf. page 16, 



24* 



