680 Mr. G. A. Schott on Radiation from Moving Systems 



It follows that a system of electrons in uniform translation 

 to the present approximation emits only waves due to dis- 

 turbances other than those produced by translation. These 

 disturbances are modified by the translation to a certain 

 extent, in accordance with the results of §§ 15-17. In 

 addition their frequencies are altered by second-order amounts 

 on account of changes in the structure of the system, and 

 those of the corresponding emitted waves by first-order 

 amounts representing the Doppler effect. 



§ 20. The result of § 19 is a consequence of the fact that 



a disturbance of the type A cos lcot + - — ) consists of a wave 



which travels round the ring with velocity co in the opposite 

 direction to that of rotation ; since by that rotation it is also 

 carried forward with velocity co, it remains at rest in space. 

 Obviously a disturbance of class of small frequency, and 

 one of class —1, of frequency nearly equal to co, give rise to 

 waves of small frequency in the surrounding medium. We 

 find such disturbances amongst the free vibrations of each 

 class. For stability @ must generally exceed 1/100, so that 

 tw is large compared with the frequency to light-waves ; but 

 the small frequencies referred to may be of the right order 

 of magnitude. Further, I have shown elsewhere * that 

 vibrations of classes 0,-1 are amongst those which are able 

 to produce spectrum lines of sufficient intensity to be observed. 

 These particular disturbances are precisely those which are 

 most strongly excited when the velocity and orientation of 

 the system are changing. Hence, although a system moving 

 with uniform velocity and with invariable orientation cannot 

 of itself produce spectrum lines, we have in changes of 

 velocity and orientation a possible source of such lines. 



It is beyond the scope of this paper to discuss these 

 vibrations fully, but it will be instructive to study the simplest 

 of them, namely those parallel to the axis. 



The equation of motion for axial vibrations is of the form 



f+pr+Q?=z/«, 



where P, Q are functions of the frequency p with real 

 coefficients, P 2 being small compared with Q. The corre- 

 sponding frequency equation for the free vibrations, 



y-<pP-Q = 0, 



has two complex roots, p^ p 2 j n °t very different from +&> 

 respectively, and an infinite number of roots, p 3 , £> 4 , ... very 



* Schott, he. cit. 



