196 



Mr. G. A. Schott on the Electron 



conditions, for the radiation now considered is continually 

 taking place. This is obviously impossible. 



Apart from this it has long been known that spectrum 

 lines vary very much in intensity and width with the con- 

 ditions under which they are produced, which in itself is a 

 sufficient proof that their energy is derived from external 

 sources. 



§ 7. We shall now consider the disturbances possible in a 

 ring of n equidistant electrons in uniform circular motion, 

 and Ihe vibrations which it emits in consequence. The 

 kinematics has been very fully treated by Maxwell*. 



Let (£, 7?, £) be the displacements from steady motion of 

 the ith electron, with reference to the moving axes of § 5. 



We may write 

 (£?,?)= (A, B^- 



-la. 



C) exp \-Kt + \/ -lUt-k~\^ 



where real parts only are to be taken. 



As we shall see, the axial vibration (?) is independent of 

 the orbital vibration (f, rj) ; hence, A, B, C, a may all be 

 taken to be real, k is a small quantity representing the 

 damping ; this we shall usually neglect, q is the relative 

 frequency, as observed by an observer moving with the mean 

 velocity co of the ring. The electrons at any instant lie on a 

 curve of k waves, with 2k nodes and loops; these waves 

 travel round the ring with angular velocity q/k forward, 

 their velocity relative to a stationary observer being qjk + co. 

 Hence for the stationary observer the frequency is q-j-kco. 



We may without loss of generality choose k to lie 



between + ^ . We shall call k the " class." The forms of 



the vibration for the chief classes are shown in the figures. 



k = gives vibrations of the ring as a whole, with all the 

 electrons in the same phase. 



* .Collected Papers, vol. i. pp. 321-323. 



