Vibrations of a Dielectric Sphere. 445 



X and f is maintained. This of course confirms Larmor's 

 formula for the radiation from a vibrating electron, and the 

 force being mechanical and equal to F cos nt, the mean rate 

 of radiation is 



1 e 2 ¥ 

 3 v 



2 F 2 / ( / m' a?n 2 \ 2 a?n 2 ^ 



or merely (? 2 F 2 /3c 3 (??i 4- m') 2 since cuijc can ordinarily be 

 neglected. 



But we notice that this is the case of a perfectly con- 

 ducting electron, and we have just seen that the decrement 

 coefficient of the free vibration is of order 10 24 . There is no 

 trouble, therefore, in neglecting this Mbration in the deter- 

 mination. 



Consider now the radiation from a dielectric electron, 

 where k is of the order already found necessary. In this 

 case Walker has found, by the same method, that for the 

 whole range of periods possible to an harmonic vibration 

 falling on the electron, there are regions in which the forced 

 vibration of the electron leads to absorption of radiation, 

 separated by others in which it leads to emission. Thus for 

 a negative particle, " If m'/m is greater than 2, there is 

 emission from infinite wave-length to very far out in the 

 ultra-violet. If m'/m is less than 2, there is absorption from 

 infinite wave-length to a certain wave-length which depends 

 on the closen ss of m'/m to 2. Unless m'jm is very nearly 2, 

 it will be in the ultra-violet." 



These results are obtained by supposing that the free 

 vibrations of the electron have died away, and we have seen 

 that the decrement factor k 2 is of order 10~ 7 for any relative 

 values of electrical and Newtonian mass of the same order of 

 magnitude, the most favourable case being that in which the 

 mass is entirely electrical. Accordingly, the free vibrations 

 must not be ignored, and the expression for the radiation 

 must be greatly modified, and will probably lose its special 

 characteristics. 



A precise determination of the matter is difficult for two 

 reasons. In the first place, the distribution of the free 

 vibrations among their several periods presents analytical 

 difficulties, and moreover, on account of the smallness of the 

 damping factor, many vibrations are of equal importance 

 with the fundamental, for in the equation (10) the value of 

 p in the approximate root X= + ipic~% must be such as to 

 make the damping coefficient p 2 m' /(m + m')K- small, or in 

 other words, p must be of order k, or 10 1G at least. As the 

 successive values of p only differ by about 3, for example, 



