﻿65G Mr. Gr. H. Livens on the Electron 



correction throughout the whole analysis for the optical 

 properties or metals along the lines laid down by Wilson 

 and Jeans ; but on attempting the problem along these lines 

 I found that the fundamental differential equation on which 

 the theory is constructed turned out to be identical with the 

 equation used by Lorentz to determine the velocity distri- 

 bution function. It was therefore preferred to adopt the 

 more direct method of attack constructed on the basis of 

 certain remarks bearing on this subject in a former paper *, in 

 order to exhibit clearly the very general validity of the method . 

 In addition, the opportunity will be taken to introduce a 

 modification of an entirely different character into the general 

 theory, which has long been considered necessary in a proper 

 treatment of the subject but which has, as far as I am aware, 

 never yet been introduced. 



2. General basis of the theory. — We shall for the present 

 assume, with all previous writers on this subject, that the 

 whole of the electrical and optical properties of any metal 

 arise from the fact that there are a large number of electrons 

 in the metal free to move about in the space between the 

 atoms. The atoms and electrons will be presumed to be 

 perfectly elastic spheres, at least so far as concerns their 

 interaction in collision : we shall also presume that the 

 atoms are comparatively of such large mass that the magni- 

 tude and direction of the velocity of any atom and the 

 magnitude of the velocity of the electron are unaffected by 

 a collision between the two. 



In the absence of any external field the atoms and elec- 

 trons will be moving about in a perfectly irregular manner, 

 and the velocity distribution will thus be exactly that 

 specified by Maxwell's law, so that if N is the number of 

 free electrons per cubic centimetre of the metal, then the 

 number in the same volume with their velocity components 

 between (£, 77, f) (£ + <i£ y + drj, %+d£) is given by 



SN = N^/^3 e^d^drjd^ 

 wherein we have used 



and g for a constant connected with the mean value u m 2 of v? 

 for all the electrons by the relation 



3 



9 = 



2uJ 



* "On the Electron Theory of Metallic Conduction," Phil. Mao-. 

 March 1915. 



