﻿based on Free Electrons. 1067 



elements. When account is taken of the fact that m 2 lm l is 

 of the order 1/1800, it is clear that if the electrical forces 

 F x F 2 are on the same scale, it will be necessary to com- 

 pensate the greater mass of positive elements by a relatively 

 small radius. The attempt to do this fails on account of the 

 strong repulsive force called into play. If there is no great 

 inequality of radii then the force on the negative element 

 must be relatively small, that is we must seek a position 

 which is nearly one of equilibrium for negative but not for 

 positive elements. There is only one such position. In that 

 position the negative ring has a less radius than the positive, 

 with a difference small except for small values of the 

 number n of elements in the ring, as will appear from 

 the following argument, certainly applicable when n is 

 not small. 



Fie. 2. 



Suppose all charges on one circle as in fig. 2, and con- 

 sider the action on a negative electron N. The positive 

 pairs on either side give a diminishing series of contribu- 

 tions to a central attraction ; the repulsions due to negative 

 p;»irs form a second diminishing series. The first series is 

 the greater as representing nearer pairs. If N were now 

 placed in the middle of the chord PiP,, the most important 

 term of the first series would be out of action, and the 

 second series representing repulsion would be the greater. 

 Between these two positions of N is the position souoht, 

 which for n great is found to show a difference in radii 

 about 2/3 of the sagitta of the arc PiP^ Rough values for 

 the ratio of radii are 1*73 for n = 2, 1*19 for ?i = 4, 1*05 

 for n = S; the departure from unity already small and 

 pointing to the asvmptotic law, variation as n~ 2 . 



3Z2 



