﻿44 Dr. H. Stanley Allen on the 



of motion of the electron in its circular path, in the one the 

 mechanical force due to the movement in the magnetic field 

 is directed towards the centre, in the other away from it. 

 These may both be included in the formula by supposing 

 that M may be either positive or negative, the positive sign 

 being taken when the mechanical force is directed towards 

 the centre. 



This equation is not in itself sufficient to specify the 

 motion completely. " There is but one equation of motion, 

 the radial one, while there are two independent variables, 

 the speed and the radius of the orbit" (Schott, 'Electro- 

 magnetic Radiation'). In order to obtain a second equation 

 we assume that the angular momentum can be expressed in 

 the form 



mr 2 a) — Th,"27T, (2) 



where h is Planck's constant, and t is a coefficient whose 

 value is not for the present specified. 



On eliminating r by means of equations (1) and (2) we 

 obtain a quadratic equation for o>, which may be written 



(Ma> + E) 2 =Aa>, 



or 



MV-«(A-iMB) + E 8 =0, . . . (3) 



1 A fJl 



where A— - — = — » . 

 bir me" 



Remembering that M may be either positive or negative, 

 we see that there are in general four possible values for w and 

 four corresponding frequencies v, since co = 2itv. 



If we divide equation (1) by the square of equation (2) 



we obtain the value of - in the form 

 r 



1 4ir 2 m , r „ . 



or approximately, when the effect of the magnetic field is 

 neglected, 



1 -iir-niEe . 



r = r 2 A 2 W 



As in Bohr's theory we consider next the work that is 

 required to move the electron from its orbit to a position of 

 rest at infinity. Denoting this quantity by W, we find, 

 assuming the mass of the core large and the effect of other 

 electrons negligible, 



w Be 1 2 o ,_. 



W = -- k) mrw- (5) 



