Theory of Metallic Conduction. 113 



I£ the positions of the velocity points are referred to axes 

 of coordinates parallel to those that have been chosen in the 

 metal itself, the coordinates of a velocity point are equal to 

 the components (f, 77, f) of the velocity of the corresponding 

 electron. 



Let dv be an element of volume in the diagram, situated 

 at the point (£, 77, f), so small that we may neglect the 

 changes of (£, 77, £) from one of its points to another and yet 

 so large that it contains a great number of velocity points. 

 This number may then be reckoned to be proportional to dv. 

 Eepresenting it by 



/(£ V, ®dv 

 per unit volume of the metal, we may say that, from a 

 statistical point of view, the function /'determines the motion 

 of the swarm of electrons. 



This function f is determined by an equation that is to be 

 regarded as the fundamental formula of the theory, and 

 which we may proceed to establish on the general assumption 

 that the electrons are subjected at the instant t to a force 

 which imparts to them an acceleration whose components in 

 the three principal directions (X, Y, Z) are the same for all 

 the electrons in one of the groups considered. 



Let us fix our attention on the electrons lying, at the 

 time t, in an element of volume dY of the metal, and having 

 their velocity points in the element dv of the diagram. It* 

 there were no encounWs either with other electrons or 

 with metallic atoms, th se electrons would be found at the 

 time t + dt in an element dY' of the metal equal to dV, and 

 lying at the point (.v-\- gdt, y + r)dt, z-j-^dt). At the same 

 time their velocity points would have been displaced to 

 an element dv' equal to dv and situated at the point 

 (£+X<ft, r} + Ydt, ^-\-Zdt), of the diagram, so that we 

 should have 



/(| + Xdt, 77 + Ydt, f + Z dt, x + gdt, y + v dt, z -f Jtff, t + dt)dY'dv [ 



=/(£ v, £ *, y, *, t)dYdv. 



The impacts which take place during the interval of time 

 considered require us to modify this equation. The number 

 of electrons constituting at the time £-f-rf£, the group speci- 

 fied by dY' and dv' is no longer equal to the number of: 

 those which, at the time t, belonged to the group (dY, </r), 

 the latter number having to be diminished by the number of 

 impacts which the group of electrons under consideration 

 undergoes during the time dt, and increased by the number 

 of the impacts by which an electron, originally not belonging 



Phil. May. S. 6, Vol. 30. No. 175. July 1915. I 



