40 Lord Rayleigh on the Incidence of Aerial 



so that, at a distance r great in comparison with X, yjr becomes,, 



^_ _ *!5 (_EJ>? (o'-o-ya+b) m ikr (70) 



Y ~~ ir \2ikr) 2{a'a+<rb) r >'" { } 



T being written for irab. The complete solution for a great 

 distance is given by addition of (66) and (70), and corresponds 

 to c}> = e ikx . 



In the case of circular section (b = a) we have altogether * 



which may be realized as in (67). If the material be un- 

 yielding, the corresponding result is obtained by making 

 m' ' = oo ; a' = co in (71). The realized value is then f 



2tt . ira? /I x\ 2ir , TT , „ 



*=~-^r\i + i) aM T^ t - r -^- ■ (72 > 



In general, if the material be unyielding, we get from 

 (66), (70) 



*=-^-(^) s (l + ^£). . . (73) 



The most interesting case of a difference between a and b is 

 when one of them vanishes, so that the cylinder reduces to an 

 infinitely thin blade. If b = 0, i/r vanishes as to both its 

 parts; but if a = 0, although the term of zero order vanishes, 

 that of the first order remains finite, and we have 



+— ^Hse)*?' • • • («) 



in agreement with the value formerly obtained J. 



It remains to consider the extreme case which arises when 

 m'=0. The term of zero order in circular harmonics, as 

 given in (66), then becomes infinite, and that of the first 

 order (70) is relatively negligible. The condition to be 

 satisfied at the surface of the obstacle is now the evan- 

 escence of the total potential (<^ + ^), in which <f> = l. 



It will conduce to clearness to take first the case of the 

 circular cylinder (a). By (62), (63) the surface condition is 



8 {7 + log(i**a).} + l=0 (75) 



* ' Theory of Sound,' § 343. 

 f Loc. cit. equation (17). 



\ Phil. Mag. April 1897, p. 271. The primary waves are there sup- 

 posed to travel in the direction of -\-x, but here in the direction of —x. 



