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



stationary, the more general case being reserved for separate 

 treatment in connexion with a discussion of the magneto- 

 optical properties of metals which is in course of preparation. 



3. The instantaneous velocity distribution under the combined 

 action of an electric and magnetic field. 



For the sake of simplicity we shall first assume that the 

 magnetic force is directed along the #-axis of coordinates in 

 a definitely chosen rectangular system of reference, and is 

 of uniform intensity H throughout the metal. The electric 

 field is assumed also to be of uniform intensity E, but for 

 the sake of generality we need not specify its direction 

 beyond stating that its components in the three principal 

 directions are (E x , E , E^). The motion of the typical 

 electron while traversing one of its free paths is thus given 

 by the following equations: 



m§i=«E, 



dt x * 



dr) x -^ <?H 

 dK\ ^ eH 



where m is the mass of an electron and e the charge on it. 

 The last two of these equations may be written in the form 



d* Vl 9 , «vE, 



d%_ o, evE 2 



dt* 



where we have used 



eU 



mc J 



and the solutions are easily obtained in the form 

 ^=7? cos vti — ? sin vh + — \ E y sin vt x + E^(l — cos vt x ) i , 



? ! = f cos vh + v sin vt x + ~ j E y (1 — cos vt x ) — E, sin vt t i , 



77, f being the respective values of t^ and f x at the time 

 ^ = 0, which in reality corresponds to the instant t. 



The accelerations of (f, rj, f) are then directly obtained by 

 differentiation in a form suitable for substitution in the 

 function %. It must, however, be noticed that the part of 



