﻿178 Mr. G. H. Livens on the 



effect impressed on an electron by the field daring its motion 

 before a collision is obliterated by the collision. During a 

 unit of time this electron traverses a large number of free 

 paths, this number being given by 



r 



where l m is the mean length of a path. Of this number we 

 know that there are a number 



whose lengths lie between I and l + dl. These contribute to 

 the above sum an amount 



^U-^e-^dl + e^'e'^l. 



Integrating this expression from to co and noticing that 

 £ is not a function of I, we find the whole contribution to the 

 sum S due to one electron in the form 



e 2 Wl 



Zmr 



(l-g) + «Ef. 



It is now clear that the mean free path l m which was merely 

 introduced in a general manner (much on the lines adopted 

 by Lorentz in his book ' The Theory of Electrons,' page 282, 

 note 36) can be assumed to have its undisturbed value, which 

 it assumes in the absence of an electric field *. 



This last expression must be summed over all the elections 

 in a unit volume. If, as above, we assume that each colli- 

 sion destroys the effects of the electric field, then we may 

 assume that the distribution of the initial velocities among 

 the electrons is that expressed by Maxwell's law ; in other 

 words the number of electrons per unit volume with their 

 velocities between the limits (f, 77, f; and (f + ^£, v + dv? 

 £+<*$) is 



V 



</\ 



wherein N is the total number of electrons per unit volume 



* Some doubt may be expressed as to the general validity of the argu- 

 ment just repeated, but I think, on due consideration, it will be difficult 

 to replace it by any other. Besides, the argument used by Lorentz to 

 deduce the law of distribution of the lengths of path is probably inde- 

 pendent of the action of the field, if lm is propavly interpreted. 



