88 Lord Kayleigh on the 



or. on substitution for b in terms of h from (23), (24), 



Vy l + P/ 1 -K^(^AK-')-^|(A A / .-')=0, (29) 

 and 



VVk+«Vi-K^(* i AK-)-0, (30) 



v%1 + P/ !l + K(£ + |)(/ i0 AK-) 



+a^^(w^-«)-a . . . (3i) 



The solution of (29) is 



-iW^i (hAK ' i)dxdi,dz ' ■ ■ (32) 



where r, equal to s/ {(a — a') 2 + (/3 — yY + iy—z) 2 }, is the dis- 

 tance of the element of volume dx dy dz from the point «, /3, 7 

 at which /i is to be estimated. 



In applying (32) to the calculation of a secondary wave at 

 a distance from the region of disturbance, we may conveniently 

 integrate it by parts. Thus, 



From the general value of r, - 



d /e' ikr \ y-ze- ikr (l+ikr) 



dz\ r J r r 2 ' ^ ' 



d? (e- ikr \ u—x y—ze-^jo + oikr—Pr 2 ) ( 



dxdz\ r J r r r s ^ ' 



If r be sufficiently great in comparison with X, only the high- 

 est power of kr in the above expressions need be retained ; 

 and if r be also great in comparison with the dimensions of 

 the region of disturbance, supposed to be situated about the 

 origin of coordinates, (*—x)/r &c. may be replaced by et/r 

 &c. Thus, 



d fe~ ikr \_yike- ikr 



dz\ r ) r r 



d? fe~ ikr \ _ «7 k 2 e~ ik ' 

 dzdx 



1 (e~ ikr \ _ 

 lx\ r )~ 



