﻿based on Free Electrons. 1101 



terms linear in p must be remembered in respect to the 

 smaller values of p corresponding toj/H-2 small; and even 

 at the other extreme the approximation is sensibly less trust- 

 worthy because the values of p are only 1/50 of those for 

 the type P 2 . 



§ 32. For a central core -\-ne and a ring of n electrons 

 the equations of oscillation are 



m,pWae=-r?,'^Z^ ) , . . . (.44) 



the two latter being simplified as to the right-hand member 

 by supposing n great, and as to the left-hand by supposing 

 p great. A specimen equation for axial motion when only 

 next neighbours are taken into account is 



/87r\N> 2 8tt 3 \ 



( . w i + - 2 -2jz 2 + z 1 + z 3 = 0, 



- . , . . 8?r 3 N c ;>» 8tt 3 

 wnicn oives to H =- a range or values trom U to 4. 



Thus for higher values of rc, ^> 2 is necessarily positive for 

 some part of the range with a transition taking place when 

 2tt 3 —ii 2 . This gives to n a value just under 8 — a value which 

 hardly justifies the approximation, and so leaves the point of 

 transition uncertain. 



T7 XT. I- 1 J- 1 l 8tT 3 N c W 2 8tT S , 



h or the radial displacement, ^ -,- ranges trom 



L n 6 n" 



to 4, and all values of p 2 are positive. In the tangential 

 displacement the form taken by the sequence equation is 



-4tt 3 N^ 



, . 4tt 3 N c w 2 „ . , n 



d makes ~— range trom —4 to u. 



an 



n 



Lastly, if the method of next neighbour is applied to the 

 case of n electrons in one ring with a different core number, 



say m e , then ^- Q — + ^ _ mc has values from to 4, N' being 

 w n 6 



m c — \c n . Complete axial stability then demands 27r 3 m c >n 3 . 



This points to a much more rapid increase of m c with increase 



of n than the special method referred to above (§ 20). But 



the oscillation method has various features of approximation, 



