786 Prof. J. H. Jeans on the 



Let ns take an element of volume dv which contains a 

 great number of electrons N dv, and let us suppose for the 

 present that the linear dimensions of dv are small compared 

 with the wave-length of the light with which we are concerned. 

 Let these electrons have velocities of components 



u, v, ?t» ; u' , v', iv' ... , 



and let the mean of these components, as before, be u , v , w . 

 Then we can replace the radiation from these electrons by 

 the radiation from an electron of charge N<? dv having velocity 

 components u Q , v , iv . If / is the acceleration of this electron, 

 the radiation from it in time t is * 



The components of the total current i in the element dv 

 are 



~Ne dv u Qi ~Ne dv v , ~Ne dv iv , 

 so that we have 



at 7 /• di 



and the radiation in time t is 



3Vj \dt) 



2 

 ~ \ dt. 



We can express — in a Fourier- series in the form 



dt 

 where 



- l (Ap cos pt + Bp sin pi) dp, 



p-0 



C di 



A p=) dt co ^ )tdt ^ 



(27) 



* Here V is the velocity in the medium, and \i is the magnetic per- 

 meability. The result is easily obtained by modifying Larmor's calcu- 

 lation ('^Ether and Matter,' p. 227). 



The radiation is calculated by integrating over a sphere of large radius, 

 so that we must imagine all absorbing and dispersing electrons removed 

 from inside this sphere. Thus V is not the velocity in the actual medium, 

 but in the medium when freed from absorption and dispersion. It is 

 given by Y 2 = C 2 /K/x, and becomes identical with V 2 , the actual velocity 

 of waves in the medium, when p = 0. 



