﻿426 Mr. G-. H. Livens on the 



between (f, ^, f) and (f + df, *? + d^ ?+df). In tnis ex ~ 

 pression N represents the total number of free electrons per 

 unit volume and 



^ is a constant connected with the mean value u m 2 of the 

 square of the resultant velocity of each electron by the 

 relation 



iutJ 



If a uniform field of strength E is brought into play 

 parallel to the d'-axis of coordinates, this velocity distribu- 

 tion will be immediately altered. Each electron will 

 acquire momentum parallel to the same axis at a rate eE 

 (e being the charge on it), but this gain will be held in check 

 by a perpetual transfer of momentum between each electron 

 and all the molecules by which it is influenced at any 

 instant. Exactly how this transfer takes place we do not 

 yet know, but some such interaction between the electrons 

 and the atoms during the encounters over and above the 

 ordinary quasi-elastic reactions seems to be necessary in 

 order to ensure the maintenance of a steady state. We can, 

 however, make good progress in the theory without assuming- 

 very much about such interactions, and two alternative 

 methods of attack have been suggested. 



Lorentz assumes that a new steady state of motion of the 

 electrons is attained, once the steady current is well estab- 

 lished, and by statistical considerations regarding the effect 

 of collisions and the electric force on the distribution he 

 finds that in such a steady state the new velocity distribution 

 may be approximately expressed by saying that 



N J'tU + -v^Wif dv d% * 



V 7r°\ mr J 



is the number of electrons per unit volume with velocity 

 components between (f, 77, £) and (^ + d^r) + drj, f+rff) : in 

 this expression m represents the mass of the typical electron 

 and l m the mean free path. This of course means that there 

 are at any instant more electrons on the average moving 



* It is here assumed that there are no thermal effects in action. 

 Lorentz's theory is, however, sufficiently general to include these actions, 

 but they modify the distribution of velocities here quoted. The present 

 mode of analysis is easily extended to such cases. 



