﻿428 Mr. Gr. H. Livens on the 



Of these dN electrons the number 



r7N - — 

 rfN= — e T ™dr* .... (ii) 



T m 



have been in motion since their last impact previous to the 

 instant t for a time which is between t and r-\-dr. In this 

 expression r m represents the mean value of the various 

 values of t for this particular group of electrons f . 



Now the velocity components of any one of the group of 

 electrons specified by (ii.) are at the instant t given by 



Thus all of them will have these components within the 

 limits (£', rj\ ?'), (f'+rff, v'+dv, V + dg), and we may 

 put 



rff=d£ d v '=d v , d£ = d£. 



If, therefore, we interpret (ii.) in terms of f ', 7} ', f ', r we may 

 conclude that there is the number 



eE 



SN' = SN 



- 1 . / 7 L \ ?'*■ / J t-/« ic.il u«i 



— V^i« a%difd%dT 



of the electrons per unit volume whose velocity components 

 at the instant c lie between (£', ?/, f) and (£' + */£', y' +drj , 

 £' + dt ) ') and for which the time t lies between r and t + ^t. 

 If we use 



r - = £ *■ -1- V + ? 



and if, as is usually the case, we may neglect terms in- 

 volving E 2 , this expression reduces to 



Thus on integration over all the values of t we find that the 

 total number of electrons with velocity components in the 

 specified limits at the instant t is given by 



?\ 2«E T ,.g' rfW ' ?r 



* Vide Lorentz, ' The Theory of Electrons/ p. 308. 

 t This value of r m also appears to be the mean duration of a free path 

 in the more jreneral case. 



