﻿1086 Mr. R. Hargreaves on Atomic Systems 



If we use only the terms of highest order in n, reduction 

 to a standard form is in fact attainable. The validity of the 

 reduction is well assured for oscillations of type P 2 , and as 

 here the motion of the mobile electron is primarily con- 

 cerned, it is to these it is proper to look for any tendency to 

 run away which instability would indicate. 



§ 17. The results of the asymptotic treatment are that 

 oscillations of type P 2 are real for displacements radial or 

 axial, unreal (or the terms exponential) for tangential dis- 

 placements. The axial oscillations of type Pj are also real, 

 but the approximations for motion in the plane are less 

 trustworthy : they point to radial stability and some degree 

 of tangential instability. 



The individual solution, for n = 2 gives real periods for 

 oscillations of type P 5 close to the orbital periods, and real 

 periods for oscillations of type P 2 with axial and radial 

 displacements, but a*n exponential form appears in the 

 tangential displacements. Thus, in respect to oscillations of 

 type P 2 , particular and general results agree in assigning 

 instability to tangential displacements, while giving stability 

 in other displacements. 



§18. Apart from the analysis, simple considerations 

 appear to be applicable when the number n is sufficiently 

 great to give near neighbours a dominant influence. In the 

 double ring each electron has ions as next neighbours, and 

 in steady motion describes a small circle relative to each ; 

 if some disturbance should accentuate tlie influence of one of 

 them, the relative orbit would tend to elliptic form. This 

 points to radial stability and tangential instability. On the 

 other hand, in the single ring (with core) eacli electron has 

 electrons as next neighbours, and if the influence of one is 

 exaggerated the tendency is to a hyperbolic form of orbit ; 

 which points to radial instability and tangential stability. 



The second consideration is that a position of rest for a 

 negative charge in the straight line between two positive 

 charges is unstable. In the double ring it appears that 

 neither the inertia of orbital motion nor the departure from 

 alignment is adequate to overcome this tendency to in- 

 stability. The argument applied to the single ring (with 

 core) suggests tangential stability. 



Again, in respect to axial motion the displacement in the 

 oscillation equation is associated with the inverse cube of 

 distance as coefficient. For next neighbours, therefore, we 

 are concerned with a factor varying as w 3 , while for the 

 action of the central core there is only variation as n. 



