4G Lord Rayleigh on the Incidence of Aerial 



importance so as to be comparable at a distance with the term 

 of the second order, although relatively negligible in the 

 neighbourhood of the obstacle. The factor, analogous to 

 — Xk 2 r 2 e~ ikr for the second order, is for the first order ikre~ tkr , 

 and for zero order e~ ikr . Thus, although (101) gives the 

 value of/ with sufficient completeness for the neighbourhood 

 of the obstacle, (102) may need to be supplemented by terms 

 of the first and zero orders in spherical harmonics of the 

 same importance as itself. The supplementary terms may be 

 obtained without much difficulty from those already arrived 

 at by means of the relations (93) , (94), (95) ; but the process 

 is rather cumbrous, and it seems better to avail ourselves 

 of the forms deduced by Hertz * for electric vibrations 

 radiated from a centre. 



If we write TL = Ke~ ihr /r, the solution corresponding to an 

 impressed electric force acting at the origin parallel to z is 



,_ dm d 2 u 7 _^ 2 n dm 



J-~ d^Tz> g -~djfcTz> /i -~^~ + ^ ; {lho) 



. dm . dm ~ /1A1 > 



These values evidently satisfy (92) since II does so, and 

 they harmonize with (93), (94), (95). 



In the neighbourhood of the origin, where kr is small, 

 e -ikr ma y kg identified with unity, so thatII = A/V. In this 

 case (103) may be written 



dm_ dm^ dm 



J~ dxdz' g ~ dxdz dz 2 ' 



and all that remains is to identify — dU/dz with ^ in (100). 

 Accordingly 



K' — K 

 A =-« 8 K^2K ( 105 > 



The values of/, g, h in (103) are now determined. Those 

 of a, /3, 7 are relatively negligible in the neighbourhood of 

 the origin. At a great distance we have 



,72 /,-i*rv Kd 2 e- ihr PAe~ ikr xz 



/_ A dxdz\ r J~ 



r 



dw dz r r 2 ' 



* Ausbreitung der electrischen Kraft, Leipzig, 1892, p. ]50. It may 

 be observed that the solution for the analogous but more difficult problem 

 relating to an elastic solid was given much earlier bv Stokes (Camb. 

 Trans, vol. ix. p. 1, 1849). Compare < Theory of Sound,' 2nd ed. § 378. 



