﻿Theory of the Optical Properties of Metals. 657 



When an electric field (of intensity E) is applied in any 

 direction all this alters ; the electrons will be pulled about 

 by the field in such a way that they will acquire momentum 

 at a rate of eE parallel to the direction of E. We shall 

 assume, however, that each collision between an electron 

 and an atom completely removes all effects imparted by the 

 electric field during the previous free motion, so that the law 

 of distribution of the initial velocities for the free paths being 

 pursued at any instant will be precisely that given as above 

 by Maxwell's law. It seems necessary to make some such 

 assumption as this in order to ensure, for instance, the possi- 

 bility of the existence of a steady state when the electric 

 force is uniform and constant in time ; and the present one 

 is probably the most general assumption we can make* as it 

 involves no detailed specification as to the dynamical nature 

 of the collisions, and also enjoys the comparatively wide 

 range of generality possessed by Maxwell's law itself. 



It is assumed that collisions between electrons are too 

 infrequent to be of any importance in the theory : this 

 appears to be a legitimate assumption on account of the 

 extreme smallness of the size of an electron. We shall also 

 assume for the present, that the motion of any charged atoms 

 is unaltered by the field, so that there will be no contribution 

 to the current on this account. 



3. The instantaneous velocity distribution when the electric 

 field is in action. — On the basis of the assumptions mentioned 

 in the last paragraph, it is possible to calculate the actual 

 instantaneous distribution of velocities at any instant when 

 the electrons are subject to the action of an electric force E, 

 which will be presumed to be applied parallel to the A'-axis 

 of coordinates chosen for the analyses. In fact the number 

 of electrons which started their paths with velocity com- 

 ponents between (f , rj , £ ) and (% + di; Q , rjo + ^Vo^ ?o + tf?o) 

 at limes which lie in the small interval between t = r and 

 t = (r-\-dr) previous to the instant t is 



where 



SN f 



= N vf3* '° rf &*M& 



* [April 1st, 1915.] It is, as a matter of fact, involved in the 

 assumption of hard elastic spheres for both atoms and elections. 

 f Vide Lorentz, ' The Theory of Electrons,' p, 808. 



Phil, Mag. S. 6. Vol. 29. No. 173. May 1915. 2 U 



