RECENT STATISTICAL THEORIES 723 



at interruptions must be balanced by the average gain between 

 interruptions. 



In the corpuscle-theory these interruptions are pictured as actual 

 impacts or collisions of the electrons with the atoms. Evidently, 

 if we could assume that whenever an electron hits an atom it rebounds 

 in some direction perfectly transverse to the field, then we should 

 have a mechanism in which the drift-speed of the electron up the 

 potential-gradient is annulled at every impact. This would be much 

 too artificial. But if we think of both the electrons and the atoms as 

 elastic spheres, the latter being so massive that they never budge 

 when struck, the result is in effect the same. For then, the angle 

 between the direction along which an electron approaches an atom 

 and the direction along which it flies away after collision is on the 

 average 90°. The rebound is as likely to be backward as forward; 

 the rebounding sphere retains on the average no memory of its former 

 direction of flight. This I will prove later. 



There is a difficulty, which I must not leave unmentioned, although 

 in this place I can do nothing to clear it away. In the development 

 of these ideas we shall in effect assume that at the end of each free 

 path the electron loses not only the forward drift-speed but the whole 

 of the kinetic energy which it acquired while traversing that free 

 path under the influence of the field. But if it collides with infinitely 

 massive spheres it does not lose kinetic energy at all. If it collides 

 with spheres of the mass of an atom, it loses kinetic energy, but does 

 not completely lose its drift-speed. The theory of this latter case 

 has been developed by Compton and Hertz for use in the study of 

 conduction in gases, and might be applied to the problem presented 

 by metals, but probably fits them no better than does the other 

 hypothesis. 



With this elastic-sphere model, then, the average interval between 

 impacts is the average interval during which the electron is piling up 

 drift-speed, only to lose it all at the end of the interval and be forced 

 to start afresh from scratch. Denote by /o the length of this average 

 interval. Since the acceleration of the electron is eE/m, its drift- 

 speed at the end of the period /o is (eEto/m), its average drift-speed 

 is half as great. Now I must dispel the impression that the drift-speed 

 is the whole of the speed which the electron has. On the contrary, 

 the mean speed of the thermal agitation — let me call it v — is immensely 

 larger than the small contribution which any ordinary field (indeed, 

 any not very extraordinary field) can impart to an electron over a 

 distance comparable with the distance between atoms. The field 

 must not be supposed to do more than bend very slightly the rectilinear 



