﻿1098 Mr. R. Hargreaves on Atomic Systems 



In the case of attraction the terms containing (p, 0) are 

 got by writing {2r-Y l)7r/ra for -v/r, (p/, 0/) for (p', #') and 

 summing for values of r from to n — 1, when such a sum 

 gives the terms in — U 2 /e 2 which involve (p, 0). In dealing 

 with repulsion a = a' } 2sir\n takes the place of i/r, (p s , S ) 

 replace (p', 0'), and 5 ranges from 1 to n — 1, when the sum 

 gives the terms in U 2 /e 2 which contain (p, 0). 



This exact expression is simplified when n is great, for 

 a — a' is of order n~ 2 and D of order n~ l for the near 

 neighbours whose action is most influential. 



The terms of highest order which contain a given (p, 0) 

 are then 



TJ 2 /6 2 = S{(p-p,.') 2 /2-a 2 (^-^') 2 }/D P 8 



r 



+ 2{-(p-/> s ) 2 /2 + a 2 (0-0J 2 }/D/ . (35 6) 



Thus a form of the same type as (35 a) is realized, but 

 only in virtue of the approximations used. 



On the inertia side also a greater simplicity attaches to 

 the coordinate z than to p, ; for if each has a factor e p(at , 

 p 2 z occurs in th« axial motion, (p 2 — l)p — 2pa0 and p 2 a0 + 2pp 

 in the plane motion. No sufficient simplification is attainable 

 unless the terms linear in p can be neglected, and this 

 requires p to be great. Now p 2 varies as n 3 /N and ultimately 

 as n 2 , i. e. its main term has a form n\ where % varies from 

 one root to another. The omission of linear terms is there- 

 fore permissible for large values of n so long as % does not 

 become small. This case of difficulty does not occur in 

 dealing with the type P 2 , but it appears in respect to type P 2 

 and also in the work for a single ring with core. Oscillations 

 P 2 have the additional advantage of showing an extra factor 

 m-^Jm^ in the value of p 2 which greatly improves the approxi- 

 mation. 



§ 30. Axial oscillations of type P 2 are represented by an 

 equation 



m 2 p 2 co 2 z= -e 2 {$ (z-z r ')/D*-2 (e-*)/D/} 



r s 



>or pp 2 z= - ^ {t [z-Zr) cosec 3 (2r + l)w/n 



— 2 (z -z s ) cosec 3 S7r/n}. (37) 



s 



In accordance with § 26 we may omit z r ', and the sum of 

 variables then shows a period given by 



^ 2 =-S3/SN=~n 2 T 3 /7r 3 log2 or p 2 = - {16-72n) 2 



in agreement with § 25. For other periods the coefficients 



