﻿Electron Theory of Metallic Conduction, 429 



which is practically the same as 



Now, noticing that r m only occurs in the small correction 

 term, we may replace it by the approximate value 



r = <- 

 r 



so that we deduce the law of distribution of velocities at any 

 instant in the steady state and under the action of the 

 uniform electric force E in the form 



which is precisely Lorentz's result *. 



It thus appears that under the assumptions made by 

 Drude and Thomson the average steady distribution of 

 velocities is precisely that derived by Lorentz. I think also 

 it is clear that the argument may be reversed, or, in other 

 words, that the steady distribution calculated on general 

 Grounds by Lorentz necessarily implies that the distribution 

 of initial velocities is that specified by Maxwell's law. The 

 two modes of formulation of the theory are consequently 

 ultimately identical, although they are apparently very 

 different in form. 



The mode of deduction of Lorentz's formula here suggested 

 can be extended so as to apply to the more general problem 

 with varying fields. As an example we may, for instance, 

 calculate the average distribution of velocities at any instant 

 when the perfectly irregular motion of the electrons is 

 modified by the application of a simple periodic electric 

 force, say 



E=E cosp(W') 



where t is used for the time t at which we evaluate the 

 distribution, and t' is an auxiliary time variable which is 

 measured from the instant t as origin. Under these circum- 

 stances we find, with the same notation as before, 



f'^f + ^.4sinp«-sino(e-T)} 



* * pm { M 



* pm \ -/ & 



while v' = V, f — ?• 



• In this expression l m is legitimately interpreted as the length of the 

 mean free path. 



