﻿1100 Mr. R. Hargreaves on Atomic Systems 



To the value y = — 2 corresponds in (38) 



fif + a-2/3=—2l3 or /*p 2 + a = 0, 



the root attaching to the sum of variables. The other 

 extreme gives }ip 2 = — a + 4/8, a negative number since 



u : 4/3 = 4T 3 = 4'2072 ; 

 this corresponds to a period 27r/14-6n&), the number .16*72 

 being reduced by a factor '873 or \/3'2O72/4-2072. 



The reduced equations for p agree with those for z, 

 cf. (35, a, h) ; in view of these reductions we find that the 

 periods for radial tend to differ from those of axial displace- 

 ments by amounts which diminish as n is increased. For 

 tangential displacements it appears from (35 b) that we 

 obtain exponentials with exponents \/2 times the values 

 found in the oscillations. 



§ 31. For oscillations of type P : we have to deal with 

 variables for both rings, and revert to the plan of separation 

 by odd and even indices. If we retain only the first term in 

 the repulsive series and the first in the attractive, then for 

 axial motion a specimen equation is 



m x p 2 co 2 z z = — ^3 { 2- 3 — z 2 - z± — I -(2 z z — z x — z- ) . . . } 



or p%=-^{2z,-z 2 -z,-i(2z 3 -z 1 -z 5 )...\. . (42) 

 In accordance with § 26, we write 



then taking only the first terms of these, viz. 2z 2 — z 1 ~z 3 — y 

 2z i — z 3 — z 5 — 0, we can clear (42) of variables with even 

 index and obtain 



'8Np 



3?i ; 



+ 2^3-^1-^5 = 0. . . . (43) 



As z. o K refer to consecutive ions, the method used above 

 is applicable and gives to 8N^V 3 /3n 3 + 2 a series of values 

 ranging from —2 to +2, or to NpV/w 3 values from —3/2 

 to 0. The greatest numerical value is jd/?i=*331 V — 1, or 

 wave-length 50'5 times the least wave-length in P 2 . But in 

 treating P 2 a ll terms z~z r ' were taken into account, here 

 only the first ; and a fairer factor of comparison is 49*2, got 

 by omission of factor T 3 in the first result. 



For radial displacement the limitation imposed by neglecting 



