﻿Electron Theory of Metallic Conduction. 181 



Arid we must now sum this expression over all the free paths 

 of all the electrons per unit volume per unit time. We 

 may notice, however, that since the phase e of the electric 

 force at the beginning of the path may have any value what- 

 ever a we may at once replace all factors involving e by their 

 mean values. For example, the mean value of : 



(ii.) cos^+e) 

 (iii.)cos«(£ + «) 



IS 



1 



2 ; 



1 pi 

 cose is ~cos— - 



r 



is 



1. 



9 ' 



(iv.) cos J — +e j sin 



pl \ . . 1 . pi 



1 — 9 sir 



and the others are all zero. 



We have thus to sum expressions referring to each free 

 path of the type 



* 2 Eo : 

 Imp 



s 2 sm 2 ^ \ { cos *— — 1 + - sm y - ^ • 



- 1 2r rl r r r J J 



Now, as before, let us confine our attention to one particular 

 electron which in the absence of any external field would 

 continue to move with velocity r, and which therefore will 

 begin to describe each path with this velocity. In unit time 

 this electron will describe on the average the number rjl m of 

 free paths, and of this number there are 



y- 2 e ' ell 



I'm 



whose length lies between I and I + ell. The contribution of 

 these terms to the above sum is therefore 



2mp 2 L 2r \ r 2 / r d r J l m 2 



- l l lm dl. 



On integration of this expression from to x> we find the 

 whole contribution by this particular electron. Noticing 

 that 



, 



*&&-'***=. 



i + 



pV 



