Theory of Metallic Conduction. 289 



electrons in such a way that the potential energy o£ the 

 electrons when at a distance r from the centre is 



m 



2\r) 



relative to the atom ; this is the most general form of law of 

 action appropriate to the present type of problem. There is 

 every reason to believe that in the simpler metals s = 2 gives 

 a sufficiently good approximation to the actual facts, but we 

 shall not make this particular assumption in the following 

 work. 



The particular case examined by Lorentz is easily deduced 

 if fju is taken equal to the radius of the atom and s made 

 infinity. 



In the more general problem now under review it will be 

 necessary to take into account a factor which was neglected 

 in the previous argument, viz., the persistence of the velocity 

 of the electron after a collision. It is well known from the 

 corresponding problems in gas theory that the velocity of an 

 electron after a collision is not entirely independent of the 

 velocity before collision; in other words, the collisions do 

 not entirely obliterate the effects of the reorganizing forces 

 in the external field produced previous to the collisions. This 

 persistence of the effects may be mathematically expressed 

 by saying that if the distribution of velocities among the 

 electrons just before collision be such that there is the 

 number £N of electrons per unit volume with their velocity 

 components in the specified range over and above the 

 number expressed by Maxwell's law, then the number in 

 the same group taken just after their respective collisions 

 is e8N, where e is a factor (< 1) which expresses the dis- 

 organizing effect of the collisions : this of course implies 

 that the expectation of the particular velocity defining the 

 group is reduced from 8N to eSN by the collisions. The 

 collisions have therefore only removed the number (1 — e)8N 

 of electrons from the group considered. 



In this way it is easily seen that 



where r m is used properly for the mean time between two 

 successive collisions of an electron. If we use 



Twt "I-e' 

 Phil. Mag. S. 6. Vol. 30. No. 176. Avg. 1915. U 



