530 Mr. G. H. Livens on the Electron Theory 



When, however, external electric and magnetic fields are 

 imposed throughout the interior o£ the metal, all this alters 

 because then the velocity o£ each electron while on its free 

 path is modified by the forces in the external fields. The 

 main problem is now to determine the new law of distribu- 

 tion of the velocities which is to replace the above Maxwellian 

 law when the effect of the fields is taken into account. Now 

 Lorentz * proves that if in the new state of the motion 



/(?, V, & * , y> *, d% drj d£ 

 denotes the number of electrons per unit of volume round 

 the point (#, y, z) with their velocity components between 

 (f, rj, f) and (f -Mf, 77-M77, ?+df) at the time t, then the 

 function /, the fundamental function of the theory, must 

 satisfy a differential equation of the form 



Bf + ~dv a? + ^d* ±v by + ^B* + -dt ~ b a ' 



wherein (X, Y, Z) denotes the acceleration components 

 impressed on the typical electron of the group by the 

 external fields and (b— a) dt; dij d^dt denotes the increase in 

 the specified group of molecules during the time dt brought 

 about by the collisions which occur during this time. 



Now as the present writer has recently emphasized, and 

 as Prof. Richardson t had previously proved in special case, 

 when the electrons are repelled from an atom in such a way 

 that its potential energy relative to that atom when at a 

 distance r from it is 



we may put 



b — a~ — 





e 



where 



1 i- 4 r°° 



— =4:7rn/jL 2 u s\ cos 2 6adu } 



Tm Jo 



6 being defined as a function of a by the equation 

 r x ° dx 



* Vide ' The Theory of Electrons,' p. 266. 

 t See Phil. Mag. April 1912. Richards- 

 steady electric fields only by the method of Lorentz 



t See Phil. Mag. April 1912. Richardson works out the case for 

 "as onh 



