﻿1102 Mr. R. Hargreaves on Atomic Systems 



and the real conditions probably lie between the limits 

 suggested by the two results ; and in particular the case 

 m e =n = 8 probably lies within the margin o£ axial stability. 



§ 33. In treating oscillations for the double ring the centre 

 was taken to be vacant, and for n large it is clear that the 

 effect of a unit centre in altering periods of oscillation is 

 slight. But a new period is necessarily introduced — that of 

 the central ion itself, which may be treated in conjunction 

 with the question of stability for central ion or electron 

 under axial displacement : a fundamental question to which 

 only a preliminary answer was given in Part I. 



A reference to (35 a) shows that it is possible by summation 

 of equations of motion to isolate the two sums of z coordinates 

 for ions and for electrons. The periods thus given (and one 

 of them is that of the central charge) are such as would 

 follow from using the same coordinates z 1 and z 2 for each 

 element of the separate rings, and this method is more 

 convenient for the purpose. Thus for a central ion with z 

 as axial displacement, z x and z 2 coordinates relative to the 

 ion, equations of motion are : — 



l 



in w 



hich 



mi'z = Z — nazi + n/3z 2 , 

 m 1 (z + z 1 )=Z + uz 1 —y(z l — z 2 ), f . , 

 m 2 (zo + z 2 ) = —Z^-8z 2 + ry{z 1 — z 2 );j 



a^e 2 /^ 3 , /3 = e 2 /a 2 3 = A, 7 = 2/3</> = 2tf 3 «£, 



(45) 



2cj>= 2{l + x 2 -2wcos{2r + l)>rr/n} 



■3/2 



r=Q 



y («) 



}. . (47) 



and for n great <£ = ?2 3 T 3 /7r 3 approximately. 



The external force is required in a later application. 

 Eliminating z , we have 



m 1 z 1 = (n + l)otz l —n^z 2 - y(z 1 -z 2 ) ) 



m 2 z 2 =—f3z 2 + y(z 1 — z 2 ), 

 or with z=—q 2 o) 2 z and ??i 1 &) 2 a 1 3 = Ne 2 , 



{q 2 N + n + l-2w*<l>)z 1 = (n-2(l>)xh 2 , 



(fiq 2 N - a? 3 - 2x*$)z 2 = - 2x 3 <j>z 1 , 

 leading to 

 ^4N S -5 2 N« 8 (2<^ + 1) + (n + l)tf»{20(a»-l) -1} = 0, (48 a) 



where the only approximation used is an omission of //,, and 

 N is the value proper to the case with centre. Where a 





