﻿based on Free Electrons. 1087 



Hence in the single ring with core the repulsive action of 

 next neighbours must be dominant for larger values ofw — 

 that is, the axial displacements unstable. The same argu- 

 ment suggests for the doable ring a thoroughly stable 

 position in respect to axial displacements. 



§ 19. The individual solutions found for the single ring 

 with core comprise the cases of two electrons with core 

 -f 2 or -f 3, and three electrons with core +3 or +4. Axial 

 displacements ar« here oscillatory. Radial and tangential 

 displacements both show instability — for two electrons by the 

 appearance of a pure exponential, for three electrons by a 

 complex form in which the real exponential has either sign. 



The asymptotic solution gives axial instability ; the 



approximations tor plane motion are less trustworthy, but, 



point to tangential stability and some degree of radial 



instability. The general conclusion for a single ring of n 



. . . . 



electrons with point-core -\-ne is that radial instability 



always exists, that axial instability accompanies increase 



of /«, while tangential instability disappears with increase 



of n. 



§ 20. There is a marked difference in the character of the 

 instability attaching to the two schemes. Consider first the 

 single ring with core. With n electrons and a core +n, 

 there is axial instability when n is sufficiently great. This 

 type of instability is taken to be fatal, and the question arises 

 whether relief can be found by distributing the electrons in 

 several rings. To obtain a first clue a less exact method of 

 calculation was followed, in which one electron is displaced 

 from the plane containing other electrons and the core — that 

 is, the mutual displacements of these elements were ignored. 

 The condition obtained is probably less stringent than the 

 real conditions. 



On this basis it appears that up to ?z = 9 a core -i- n will 

 give axial stability to n electrons ; but for greater values 

 of n (number ot electrons) stability demands a rapidly 

 increasing excess of the core number. A tabulated result of 

 calculations made by this simplified method suggests that 

 relief can be found by distributing the electrons in a 

 succession of rings, which need not exceed but must reach 

 seven in dealing with the greatest value of n required. 

 These figures may be modified by more exact treatment, 

 cf. § 32. The effect of distribution in several rings on radial 

 and tangential displacements has not been examined. 



§ 21. Consider now with respect to the double ring the 

 question of tangential instability, the only type of which 



