*+*ff'+*f'f")dt 



t 



400 Mr. GL H. Livens on the Flux of 



which gives for the amount or! energy between the spheres 

 of radii r and r + 6> the total T r Sr, where 



= ^('' 2 /" 2 + 2/ ,2 + ^J/"^«). 



On Macdonald's form of the theory this expression turns 

 out to be 



T,=i('- 2 /7'" + 2/' 3 + 2r/'/"); 

 whilst on Poynting's theory it is simply 



The distribution of energy on the various theories is 

 therefore essentially different. At a great distance from 

 the vibrator — that is, in the purely radiation part of the 

 field — both Poynting's theory and the general form of the 

 new theory give the same energy density, in a form which 

 corresponds to a value of T r given by 



±r — 3/ 

 But on Macdonald's theory the density is such that 



T r = Iff", 



which, however, agrees on the average with the former 

 expression in the case of simple harmonic oscillation ; in 

 other cases it is essentially different, and if the acceleration 

 of the moment of the doublet is constant 



/'" = o, 



and the density of the kinetic energy in the distant field is 

 then of a smaller order of magnitude than in either of the 

 other cases, although, as we shall see later, the outward flux 

 of energy is more than in Poynting's case. 



Up near the radiator the main part of the energy density 

 on either form of the new theory is twice what it is on 

 Poynting's theory, but on Alacdonald's own form it is 

 not otherwise essentially different. On the more general 

 form of the theory, however, the generally less important 



