288 Mr. G. H. Livens on the Electron 



solution of a differential equation, which he writes in the 

 form 



d£ d*? d? % ^os 'dy *^z dt 



Herein the function / is defined so that the number of 

 electrons in the specified group is 



/(? , v, & «, y, *i ^f • ^ • <*?; 



(X, Y, Z) are the component accelerations produced on the 

 typical electron of the group by the external fields and the 

 same for all of the electrons in the group ; the difference 

 (b— a) is such that (b—a)dl; drj d£ represents the increase 

 per unit volume of the number of electrons in the specified 

 group brought about per unit time by the collisions of these 

 electrons with the atoms. 



In the particular case examined by Lorentz when both the 

 electrons and atoms are assumed to be hard elastic spheres, 

 the atoms being rigidly fixed, the number (b — a) may be 

 simply replaced by 



fcrf 



where f is the particular value of / which expresses Max- 

 well's law of distribution of the electronic motions and T m is 

 the mean time between two successive collisions of an elec- 

 tron of the specified group. The equation for / is in this 

 case 



of which the general solution may be immediately written 

 down. 



It was thought at the time when the previous paper of 

 this series was written, that this equation was perfectly 

 general and independent of the particular dynamical nature 

 of the interaction between an electron and an atom on 

 collision. In a certain sense this statement is true, although 

 in the general case it is necessary to specify very carefully 

 the appropriate value of i m . The object of the present paper 

 is to formulate explicitly the problem for the more general 

 case and thereby to determine the appropriate form for 

 r m to be used with the equation. 



We may generally assume that the atoms are rigid 

 spherical nuclei fixed in the metal, which act on the free 



