550 Mr. G. H. Livens on the Electron 



2. The bases of Lor entz's theory. 



According to Lorentz* the whole theory turns on the 

 evaluation of a function / which determines the statistical 

 distribution of the motions among the electrons in the metal. 

 This function is such that in a small volume element round 

 the point whose coordinates referred to a definitely chosen 

 system of rectangular axes are (#, y, z), the number of 

 electrons per unit volume at the time t with their velocity 

 components between (f, 77, f) and (f-f-df, v + dv> ?+^?) 1S 



/(£ V, K> ®>y, z , t)dV, 

 where dY = dl; drj d£. 



If the electrons at the point under consideration are subject 

 to action by external fields, the results of which may be 

 specified by the accelerations (X, Y, Z) which they impose 

 on the typical molecule, the function / is shown to satisfy a 

 differential equation of the type 



wherein (b — a)dVdt denotes the increase in the specified 

 group of electrons during the next succeeding small interval 

 dt brought about by the collisions taking place in this 

 interval. 



The main difficulty experienced in the application of this 

 equation lies in the determination of (b — a). The following- 

 remarks may, however, simplify the general problem in this 

 respect. 



If, as we shall first assume, we may neglect the persistence 

 of the velocity of an electron after collision, it is clear that 

 the velocities of any electron at two instants between which 

 it has encountered an atom are wholly independent of one 

 another. This means that the distribution of the velocities 

 among any group of electrons taken each immediately after 

 its next collision succeeding the instant t, is entirely inde- 

 pendent of the distribution at the instant t, and will in 

 general be different from it unless indeed the initial distri- 

 bution is that specified by Maxwell's law, which is specially 

 chosen so as to be unaltered by the collisions. A direct 

 inference from this point of view is that the distribution of 

 velocities among any group of electrons each taken imme- 

 diately after its next collision after the instant t, will in fact 

 be precisely that specified by Maxwell's law, and is there- 

 fore the same independently of the state of the motion that 

 * See, for example, « The Theory of Electrons/ pp. 266-273. 



