﻿176 Mr. G. H. Livens on the 



The equations of motion of the typical electron during its 

 free motion between two collisions while under the action of 

 the electric force is 



dt dt dt 



where m denotes the inertia mass of the electron and e the 

 charge on it. Thus 



?*= — +fe ?*=*> &=6 



(?» ^ ?) being the velocity at the instant of beginning the 

 free motion from which the time is also measured. The work 

 done by the electric field on this electron during the whole 

 of the time between two impacts, an interval of length r, is 

 thus 



Wp 2 T 2 



H 



and we now require the sum of the quantities of this type 

 corresponding to all the free paths of all the electrons per 

 unit of volume covered during the unit interval of time. 

 Denoting this sum by S we find that the total amount of 

 heat developed per unit time per unit volume is 



/EV t 2 \ 



Thomson and Wilson both proceed by making a statement * 

 which is equivalent in the present notation to saying that 



S(«E£t) = 0, 



their reason being presumably that since positive and nega- 

 tive values of f are equally probable there will be equal 

 positive and negative terms in the sum which w T ill thus on 

 the whole be zero. This statement, however, does not appear 

 to be quite correct, since, granted exactly identical conditions 

 for electrons with the component velocity f, the value of t is 

 less when f is positive than when it is negative by an amount 

 of the order <?Er, so that there must in any case be a residue 



* It is perhaps only fair to add that this statement, in the form I give 

 it, probably never occurred to either author, since in the particular case 

 they examine they have another plausible reason for neglecting the 

 corresponding term of the sum. 



