﻿1104 Atomic Systems based on Free Electrons. 



This gives 



{y(/3-a)-a^}(z 2 -z 1 )=:{(7i + l) c~np}Mf=M.foc 



approximately for n great, and so 



Mf= y (x*-l)(z 2 -z i ) = ?>fr(z 2 -z 1 ), 



or Z = 3w£y(s,— cO. 



If now we write Z which is the force on one ion as e 2 /d 2 y 

 with the action of one atom on another at interatomic 

 distance in view, and apply numerical values, we get 



z 2 —z x _ 1-512 a 2 

 ' X d 2 *' 



a n 



and a comparison with — 2 = — ^- shows that the relative 



1 a n l 



displacement of the rings is small compared with the differ- 

 ence of radii. The value of z x or z 2 is much greater than 

 their difference, viz. M/= — (/3 — *)z\ or Z= — 3nt;az l9 with 

 a numerical result z 1 /a = — na 2 /9-75d 2 . On this feature that 

 the displacement of the central ion relative to either ring is 

 on a much greater scale than their mutual displacement, 

 is based the remark on electrolytic conduction in § 5, Part I. 

 If Z is not a constant but a periodic force of period 

 27r/Qft), the method used above will be found to give 



£ 2 a-^ 1 [ A 6Q 4 N 2 -Q 2 N^(2^+ 1) 



4- (7z+l).r 3 {2(/)(^-l) + l}]r=-^(2^-n^Z, 

 or also y (50) 



m 2 z l( om(Q 2 -q 2 ) (Q 2 -a 12 ) = -^(2<j>-n)Z, 



m 2 e 2ft , 2 N(Q 2 -^ 2 )(Q 2 - 9 /2 )--(2^-n-l-Q 2 N)Z^ 



with q and q' as in § 33. When there is no central charge 

 the formulae are 



m 



2-2 



o) 2 (Q 2 — q 2 ) = Z 1 and m 1 zi + m 2 z 2 = 0. 



The method could no doubt be applied to examine the 

 scale of displacement of satellites, but not without a sensible 

 complication of the equations. 



In bringing this long task to a close I wish to acknowledge 

 the kindness of Sir Joseph Larmor in reading earlier sketches. 

 of this paper, and making various comments, criticism's and 



