840 Prof. H. A. Wilson on the Electron Theory 



Let a cos pt denote the electric force so that mdvjdt 

 = ae cos pt, where m is the mass, e the charge of an electron, 

 and v its velocity component parallel to the electric force. 

 Let t denote the time at the beginning of a free path and 

 f + T that at the end. Then 



m(v r — v )= — {sinp(£ + T)— sinp^}, 



lb 



or . 2ae p /a . . pr 



m {v T - v o)=— cos|-(2* + T)sin^ . 



Therefore 



4m(^-V) = i(57 COs2 1 ( 2t » + T ) sin2 f 

 + 2» -oos|(2t +T)sin^. 



P Lt Z 



If now S denotes a summatiou for all the free paths 

 described by one electron in one second, we get 



Sim(rv*-V) = S^) 2 sin«f, 



because the mean value of cos 2 x'\s\ and of cos x 0, and we 

 may make the addition in groups such that t is constant in 

 each group. 



Suppose that the velocity V of a particular electron remains 

 constant, then the number of free paths of length between I 



and l + dl in one second is N e ' lm dl\l 2 m , where l m denotes the 

 mean free path. Also / = Vt. Hence 



2mV(l+p 2 ^/V 2 )* 



The number of electrons for which V is between V and 

 V + dY is, according to Maxwell's law, 



where ^ = 3/2V and V is the mean value of V 2 . Hence 



