34 



From (32) and (34) we see that the motion in the present case may be con- 

 sidered as composed of a number of linear harmonic vibrations parallel to the axis 

 of symmetry and of frequencies equal to the absolute values of (tj Wj -[- "2 ^2)1 

 together with a number of circular harmonic motions round this axis of frequencies 

 equal to the absolute values of (tj (Wj + 73 Wj + '"3) and possessing the same direc- 

 tion of rotation as that of the moving particle or the opposite if the latter expres- 

 sion is positive or negative respectively. According to ordinary electrodynamics 

 the radiation from the system would therefore consist of a number of components 

 of frequency | r^ fUj + r^ cu^, [ polarised parallel to the axis of symmetry, and a number 

 of components of frequencies [tj^w^^ -oWa + Wgi and of circular polarisation round 

 this axis (when viewed in the direction of the axis). On the present theory we 

 shall consequently expect that in this case only two kinds of transitions between 

 the stationary states given by (22) will be possible. In both of these n^ and n^ 

 may vary by an arbitrary number of units, but in the first kind of transition, which 

 will give rise to a radiation polarised parallel to the axis of the system, n^ will 

 remain unchanged, while in the second kind of transition 72, will decrease or in- 

 crease by one unit and the emitted radiation will be circularly polarised round 

 the axis in the same direction as or the opposite of that of the rotation of the par- 

 ticle respectively. 



In the next Part we shall see that these conclusions are supported in an 

 instructive manner by the experiments on the effects of electric and magnetic fields 

 on the hydrogen spectrum. In connection with the discussion of the general theory, 

 however, it may be of interest to show that the formal analogy between the ordinary 

 theory of radiation and the theory based on (1) and (22), in case of systems pos- 

 sessing an axis of symmetry, can be traced not only with respect to frequency 

 relations but also bj' considerations of conservation of angular momentum. 

 For a conditionally periodic system possessing an axis of symmetry the angular 

 momentum round this axis is, with the above choice of coordinates, according to 



(22) equal to „-- = n„ z-—. If therefore, as assumed above for a transition corres- 



^ 2 TT ■* 2;r 



ponding to an emission of linearly polarised light, n^ is unaltered, it means that 

 the angular momentum of the system remains unchanged, while if n^ alters by 

 one unit, as assumed for a transition corresponding to an emission of circularly 



polarised light, the angular momentum will be altered bv s— • Now it is easily 



seen that the ratio between this amount of angular momentum and the amount of 

 energy hv emitted during the transition is just equal to the ratio between the amount 

 of angular momentum and energy possessed by the radiation which according to 

 ordinary electrodynamics would be emitted by an electron rotating in a circular 

 orbit in a central field of force. In fact, if a is the radius of the orbit, v the fre- 

 quency of revolution and F the force of reaction due to the electromagnetic field 

 of the radiation, the amount of energy and of angular momentum round an 

 axis through the centre of the field perpendicular to the plane of the orbit, lost 



