﻿based on Free Electrons. 1073 



multiplicity of options is to be regarded as a valuable asset 

 or as a source of embarrassment may be left to the reader to 

 decide. The loss of such large freedom of choice is the price 

 paid for the nuclear cement provided. The nucleus is of an 

 entirely different order, one of mass rather than of charge, 

 and it is endowed with orbital motion and possibilities of 

 internal oscillation. The whole structure — rings, central 

 ion, and satellites — does not admit of any but a quite small 

 residual charge. Cf. § 23. 



We proceed to the mathematical theory on which this 

 general account is based, so far as it relates to steady motion. 

 A second part contains an investigation of the natural oscil- 

 lations and of the associated question of stability, considered 

 with reference to the present scheme and also that of a 

 multiple core. 



Mathematical Theory. 



§ 8. Let there be n charges e each with mass m l on a 

 circle of radius a 1? and n charges — e each with mass m 2 on 

 a circle of radius <2 2 , disposed as above. 



For uniform motion in a circle we have 



m l w 2 a 1 — F l5 m 2 co 2 a 2 = ¥ 2 (1) 



When F X F 2 are expressed in terms of the radii, the problem 

 is to determine the ratio sc = a 1 : a 2 so as to satisfy 



F 2 /a 2 =m 2 /m 1 xF 1 /a 1 (2) 



With ratio known the forces can be expressed in terms of 

 one linear magnitude. Since m 2 : mj = fi is of order 1 : 1800 

 a first approximation is got by determining x from F 2 = 0, 

 and then finding w from 



m l co 2 a 1 = Fi (3) 



with the special value substituted in F x . The central repul- 

 sive force on one charge in a circle of n charges is given by 



e 2 c n /2a 2 



—i ... . (4) 



1 



where c n =. ■§• 1) cosec S7r/n. ) 



s = l J 



The attractive force on — e due to the n charges +e of the 

 circle a x is given by 



"v 1 € 2 \a 2 — a 1 cos ( 2r + l)7r/n} 



r=o W + a 2 2 -2 ai a 2 cos (2r + lW"P* ' ' W 



