104 Lord Rayleigh : Further Applications of BesseVs 

 appears * that 



B n =( )V*, .... (14) 



71 \ irz cos aj v ; 



where p = i7r + z{cosu — {^ir — «) sin a}, . . (15) 



or when z is extremely large (a=0) 



W=(~)V i(i " + '> (16) 



At a great distance the value of (f> in (9) thus reduces to 



^(^osnff.e^^'h . . (17) 



from which finally the imaginary part may be omitted. 



When on the other hand zjn is decidedly less than unity, 

 the most important part of (13) arises from the first and 

 last integrals. We set n = z cosh /3, and then, n being very 

 great, 



D ^ = ( £ SrT^- -ie%- . . (18) 



where t = n (tanh/3— /3) (19) 



Also, the most important part of the real and imaginary 

 terms being retained, 



n ,, N /sinh/3cosh/3\£ _, , - _ 



D n(z)=-\--%^)&e-t+ie*}. . (20) 



The application is now simple. From (9) with introduc- 

 tion of an arbitrary coefficient 



l Q=kAe ikYi cos n0.-D n '(kr). .. . (21) 



If we suppose that the normal velocity of the- vibrati no- 

 cylindrical surface (r=a) is represented by e ikYt cos nO, we 

 have 



kAD K '(ht)=l, ( 22) 



and thus at distance r 



*-™«-^^j, .... (23) 

 or when r is very great 



n { 2 \ie;{W->-)-H , N 



*= cos "*fc) hwr- ■ ■ ■ (u) 



* Nicholson, B. A. Report Dublin, 1908, p. 595 ; Phil. Mag. vol. xix. 

 p. 240 (1910) ; Macdonald, Phil. Trans, vol. ccx. p. 135 (1909). 



