Theory of Metallic Conduction. 115 



nothing to modify the velocity distribution (specified by 

 Maxwell's law) during the free-path motions prior to the 

 collisions which occur in this small interval. As a matter of 

 fact, however, certain electrons which have been moving on 

 & free path for a finite time since their last collision previous 

 to any instant t and which started with velocities outside the 

 specified limits, are brought into the group by the action of 

 the external agencies and others are made to leave it in the 

 same manner, so that at the instant t the number of electrons 

 in the group is different from the number as specified by 

 Maxwell's law. Moreover, some of the electrons added to 

 the group by external agencies previous to the instant t will 

 be removed from it by collision during the next small 

 interval dt, and these must therefore be included in adtdVdv ; 

 it is, in fact, precisely these collisions that make the differ- 

 once between adYdvdt and bdVdvdt. We may therefore 

 in such a case state that the total number of electrons 

 extracted from the specified group by collisions during the 

 interval dV, or 



(a~b)dVdvdt 



<can be specified as the number of collisions which occur 

 during the same interval among those electrons in the 

 specified group at the time t, which are over and above 

 the number of electrons in the same group as would be 

 specified by Maxwell's law for the same instant. 



But in calculating the number of collisions in the small 

 interval dt we need not trouble about the state of the metal 

 varying from one point to the other, or even of the effect of 

 the external field ; we may, in fact, simply calculate the 

 number characteristic of the particular group of electrons, 

 which is the number of collisions they would undergo if they 

 started at the same instant t with the same motions, but 

 under the action of no external or internal agencies. 



Now if N denotes the number of electrons per unit 

 volume at any point in the metal, and u m 2 the mean square 

 of their velocities, the number of these electrons which 

 have their velocity components between (f, 17, f) and 

 (?+<J£, V + tty* ?+<'?) according to Maxwell's law is deter- 

 mined by 



Ae-**dv 9 

 wherein 



■and 



u 2 = f 2 + rf "+ £*j dr = d^drjd^, 



12 



