Theory of Matter and on Radiation. 201 



with a ring of n electrons, arranged as in the diagram, and 

 revolving in the direction of the arrow. 



This is not a possible state of steady motion for a ring of 

 n electrons, because 



(1) they are not equidistant; 



(2) the angular velocity and radius correspond, not to 



n electrons, but to n + 1. 



When n is at all large, the angular velocity and radius are 

 practically the same for n electrons as for w+1. The 

 deviation (2) produces slight oscillations in velocity and 

 radius about the values corresponding to n electrons; but 

 they are very small compared with the oscillations due to (1). 

 "We shall for simplicity neglect them. Thus we treat the 

 ring as one of n electrons initially displaced from the state 

 of steady motion, with zero radial displacement and zero 

 velocity. 



For the angular displacement I find 



277 



-S(-i) 



k^q' sin (qt 



2ttA 

 n ) 



n(n + l) (£ + ?') sin (for/re)' 



and therefore for the angular velocity, 



/ 9 7r/\ 



27r ^ ^iqg'cos^qt-k—) 



S(-l) 



n(n + 1) (g + q) sin (to In) 



where the summation is for all values of k between + -, 



L 



excluding zero, and q, q' are the frequencies corresponding 

 to classes k, —k. The effect of damping has been neglected, 

 so that the amplitudes are approximate only for the initial 

 stage of the disturbance. 



It is to be remarked that, as in the case of Saturn's rings 

 considered by Maxwell *, there are four frequencies for dis- 

 turbances in the plane of the ring ; for two of these the 

 tangential displacement is a large multiple of the radial 

 displacement, for the remaining two it is about twice as great ; 

 the values of q, q' belong to the first pair, because in our case 

 the radial disturbance is negligible. 



For vibrations of classes + k we have initially 



q[ n 7 



a ° ~ + !)(? + ?0 ^n (ibr/n) 



_ q n __ 7 . 



(n+l)(g + g') sin (torn) 01 ' 



* Collected Papers, vol. i. p. 321. 



