﻿164 Mr. G. H. Livens on Lorentz' } s 



Consequently the energy flux through w' belonging to the 

 interval o£ wave-lengths d\ is given by 



c 2 6iu'a s 2 a\ 



and we now want to find a s . 



From the value of E z given by equation (1) we see that 



1 x i C ' S7rt 4<l 



a s =— , Q 2 

 zirvc-y 



where the square bracket round the Vx serves to indicate the 



value of this quantity at the time t — . The sign X now 

 i j c » 



refers again to a sum taken over all the electrons in the part 

 icA of the plate. 



On integration by parts we find 



se ^ C 6 r n sirt , 



JO 



or what is the same thing 





dt. 



Now each of the integrals on the left is made up of two 

 parts,, arising respectively from the intervals between the 

 consecutive impacts of the electrons and from the intervals 

 during these impacts. If, as mentioned above, we can 

 suppose the duration of an encounter of an electron with an 

 atom to be very much smaller than the time between 

 two successive encounters of the same electron, we may 

 neglect altogether the part that corresponds to the collisions 

 and confine ourselves entirely to the part corresponding to 

 the free paths between the collisions. But while an electron 

 travels over one of these free paths, its velocity v x is constant. 

 Thus the part of the integrals in a s which, corresponds to one 

 electron and to the time during which it traverses one of its 

 free paths is therefore 



V 



'4 cosS e( t+ 7) dt > 



where t is now the instant at which this free path is com- 

 menced and t the duration of the journey along it ; but this 

 is equal to 



v x f . sir I . , r \ .sir/ r \\ 



sitt sir ( r t\ 



*26v x 

 sm 



S7T 



