of the Optical Properties of Metals. 837 



assuming that collisions do not alter the velocities of the 

 electrons in magnitude. 



In the following paper the atoms are regarded as hard 

 smooth spheres which remain at rest, while the electrons 

 are also regarded as spheres and their mutual actions are 

 neglected. These are the assumptions adopted by H. A. 

 Lorentz* in his theory of the electrical and thermal con- 

 ductivities of metals. 



Let n denote the number of atoms per c.c. in a metal and 

 N the number of negative electrons, each with a charge e. 

 Also let R be the sum of the radii of an atom and an electron. 

 Let f , 77, £ denote the velocity components of an electron and 

 V its resultant velocity. Following Jeans, I shall begin by 

 calculating the rate of increase in the momentum of the 

 electrons due to an electric force X parallel to the x axis. 

 Let SN be the number of electrons per c.c. having velocity 

 components between f, 77, £ and f+5f, 77 + S77, and £+S£\ 

 Consider a particular atom and a small cone of solid angle 

 da) with its vertex at the centre of this atom. Let f be the 

 angle between the axis of this cone and the axis x. The 

 number of electrons in the group SN which collide with the 

 atom in time dt, so that at the moment of the collision the 

 line joining the centres of the atom and electron lies inside 

 day, is 



WR?dw cos 6Vdt, 



where is the angle between V and the axis of the cone. 



The loss of momentum in the x direction at each such 

 collision is 2mV cos cos/, where m is the mass of an electron. 

 Consequently the total loss of momentum by the group SN 

 is 



2m V 2 SN R 2 n dt j* cos 2 6 cos fdco 



= 2mV 2 SNR 2 n^7rf/2Vt 



= SN ?nf V dtjlm where l m = 1/irn R 2 . 



Now let dN denote the number of electrons per c.c. which 

 have velocities between V and V + dV, and let udN = f f$N 

 so that u is the mean value of f for the electrons in the 

 group dN. Then the loss of momentum by the electrons in 

 this group is 



mVdt (*_ __ mYu dt dN 



—j — USN= j . 



Lm l l m 



* ■ Theory of Electrons/ pp. 26G-273. 



f See H. A. Lorentz, ' Theory of Electrons/ p. 272, 



