Theory of Metallic Conduction. 551 



may exist at the time t : in other words, the collisions com- 

 pletely obliterate any regularity which existed in the elec- 

 tronic motions previous to them. It follows again that the 

 number (bdVdt) of electrons which enter the specified group 

 during the small interval dt is precisely the same as the 

 number which would enter the same group if Maxwell's law 

 specified the distribution both before and after the collisions, 

 and it might therefore be calculated on this basis. The 

 number (adVdt) of electrons leaving the group in the same 

 time would then be exactly the same as (bdVdt) if there 

 were no external forces or condition gradients in the metal 

 to modify the distribution established by the collisions. In 

 the more general case, however, it is at once obvious that 

 the number 



(a—b)dVdt 



can be calculated as the number of electrons removed by 

 collision during the time dt from among the partial group 

 of electrons contained in the specified group at the instant t, 

 which is the excess of the number in this group at the 

 instant t, over and above the number in the same group as 

 specified in Maxwell's law. 



The distribution of velocities which is expressed by Max- 

 well's law may, for the present, be taken to be specified as a 

 particular case of the distribution described above, in which 



/(fc »; 5 K *i y, s> =/o (?, v, & *> y, z , *)> 



/ being a previously assigned function of known type. The 

 distribution of electronic motions expressed by this function 

 fo is the only perfectly chaotic distribution consistent with 

 the general dynamical assumptions regarding the collisions 

 between the electrons and atoms, any departure from it 

 being the result of external forces tending to organize the 

 irregularity in the motions. 



The partial group of electrons described above has then 

 the number 



SN = (/-/<>) <*V 



per unit volume with their velocity components in the small 

 range dV , and the number of them which is removed by 

 collision during the small interval dt is easily calculated by 

 a well-known argument, and is 



f^&dVdt, 



where r m is the mean time of duration of the free path 



