390 Lord Rayleicrh 07i the Work done b>/ Forces 



the limit finite and independent of z/r. If W be the work 

 done in time dt. 



d\y 



dt 



= Zi co< jjt . 'iky A cos pt : 



and by (6), (17) 



d\y 



mean — r- 

 dt 



k^Zi 



(25) 



The right-hand member of (25) is thu< the work done (on 

 the average) in unit of time. 



This result may be confirmed by a calculation of the 

 energy radiated away in unit time, for which purpose we 

 may employ the formulpe (20). (21) appKcable when r is 

 great. The energy in question is the double of the kinetic 

 energ^^ to be found in a spherical shell whose thickness 

 (?'o — 7=1) is the distance travelled by transverse waves in the 

 unit of time. viz. h. In the expression for the kinetic energy 

 the resultant (velocity)^ at any point x, y, z is by (20), (21) 

 proportional to 



7. + 



ifz"- (r^-z'^Y 





/-^. 



viz 



Also 



1 



a quantity symmetrical with respect to the axis, 

 sin- [pt — kr) is to be replaced by its mean value, 

 Thus the kinetic energy is 



the double of which is identical with (25). 



We will now form the expression for the resolved displace- 

 ment at P due to Z^ Qo^pt acting at (parallel to OZ), the 

 displacement being resolved in a direction PT in the plane 



Fie. 1. 



ZOP making an angle 6' with OP (fig. 1). The angle 

 between PT and OZ is denoted by (j>, so that ^=6 + 6'. 



