Velocity of Swiftly Moving Electrified Particles. 593 



In the electron theory it is shown that the electric force, 

 exerted on an electron at rest by a particle of charge E and 

 uniform velocity V = /3c, will be directed along the radius 

 vector from the particle to the electron and given by* 



F==£ E 1-/3" 



r 2 (l-/S 2 sin 2 to)F 



where r is the distance apart and co the angle between the 

 radius vector and the path of the particle. Let the shortest 



distance from the path to the electron be p, and let co — — 



at the time t = 0. We have then sin co=~ and r 2 =(Vt) 2 +p 2 . 



For the components of the force perpendicular and parallel 

 to the path of the swiftly moving particle we now get 



F l= =2ir and F 2 =-F 

 r r 



respectively. Introducing for r, and putting (1 — /3 2 )~$=y, 

 we get 



TT _ P e 7 E • AnA v _ yVteE 



"We see from these expressions that the force at any moment 

 is equal to that calculated on simple electrostatics, if we 

 everywhere replace the velocity Y of the swiftly moving- 

 particle by 7V, and, in calculating the component perpen- 

 dicular to the path, replace the charge E of the particle by 

 7E, while leaving it unaltered in calculating the component 

 parallel to the path. In the calculation of the correction due 

 to the high speed of the /3 rays we shall, therefore, have to 

 consider the effects of the two components separately. 



If the electron is free it will be simply seen that the velocity 

 of the electron, after a collision in which a is small compared 

 with p, will be very nearly perpendicular to the path of the 

 /3 particle. In calculating the energy transferred in this 

 case we need therefore consider only the component of the 

 force perpendicular to the path. If V is small compared with 

 c we get from (1), neglecting a compared with p, 



m V 2 p 2 

 If in this expression we introduce 7V for V and 7E for E, 



* See, for instance, 0. W. Richardson, 'The Electron Theory of 

 Matter,' p. 249, Cambridge 1914. 



Phil. Mag. S. 6. Vol. 30. No. 178. Oct. 1915. 2 Q 



