556 Mr. G. H. Livens on the Electron 



Maxwell's law. Thus, if we take any group of electrons at the 

 instant t and trace each one back for the corresponding time 

 t to its last collision, the distribution of motion among them 

 will be precisely that specified by Maxwell's law. This fact 

 combined with a knowledge of the motion of the electrons 

 during the times t, as affected by the external field, may be 

 used to obtain a form of the instantaneous velocity distri- 

 bution, and in the following manner. 



We know that if we take any group of electrons, SN in 

 number, all with the same velocity, then the number of them 

 which have travelled for a time between t and r + dr since 

 their last collision is 



--dr 

 d8N = SXe T »-. 



Now the number of electrons per unit volume which according 

 to Maxwell's law have their velocity components between 

 (£ v, f) and (f + rf£ v + d V , E+dg) ™ 



a formula which it will for the present purpose be more 

 convenient to write in the equivalent form 



F (u 2 , *, y, e 9 t) dV. 



Of these electrons the number as above which have travelled 

 for the time between t and r + dr since their last collision 

 previous to the instant t is 



r 



Now let us examine the motion of one of these electrons. 

 It starts with the velocity (£, y, f) and moves with accelera- 

 tion (X, Y, Z) for the time t: if we denote by (f,, r\ t , £*) 

 the components of its velocity at the time t, and (#*, y t , zi) 

 the coordinates of its position at this same instant, and if 

 also we introduce, as in the last paragraph, the auxiliary 

 time variable t u we shall have 



**£*i _ y dr l^ - V d &i - 7 

 ^ _Al ' ~dh l5 ~dti ~ U 



dxt x t dyt x dz t _ 



dT x -^ 7^ = ^' dt[~^ 



