﻿based on Free Electroiis. 1093 



to sums of displacements in one ring, and provides a useful 

 check on results obtained by approximative methods. The 

 periods differ widely, and the fact that displacements p x and 

 p 2 are o£ like order for one, while m^pi and m 2 p 2 are of like 

 order for the other, suggests a principle which may be 

 applied to separate the groups, and thus halve the number 

 of equations needed to determine the periods in each 

 group. 



§ 26. To explain this method, lefc#i# 2 he typical coordinates 

 respectively with positive and negative charges, and let U 2 

 stand for the second-order term in the potential energy. In 

 dealing with the first group of periods we may omit inertia 



terms with m 2 and write -^— ; 2 =0, by use of which a new 



(to 

 form ot U 2 is reached containing only variables x n and 

 accordingly the group of equations with m 1 as inertia 

 coefficient is formed. These equations give values in which 

 the ratio jjl is neglected, and we can use x and N as tabulated 

 in lieu of N^. 



In the second group, fxq 2 is finite when p> is accounted 

 small, and so q is a large number ; hence those inertia terms 

 which are linear in q may be ignored in finding a first 

 approximation. For the motion of positive charges, x x 

 appears on the inertia side with a multiplier m^W, on the 

 other side X\ and x 2 have as a multiplier e 2 /a 3 , which is of order 

 m^ 2 . Hence x 2 is of order (fx\ or x l of order fxx 2 ; and so 

 variables .i\ may be ignored on the potential side of both sets 

 of equations. The first set then gives x x in terms of x 2 with 

 a multiplier varying as q~ 2 or as ft, and the second set 

 determines a group of values of /j,q 2 . 



The separation presumes that we are content with the 

 main terms in the sense that order is determined by the 

 small number /z, and the groups present q 2 of the one and 

 fig 2 of the other as numbers on the same scale. 



§ 27. We now apply this to oscillations in the plane for 

 n = 2. It is convenient to mark with an odd index 

 coordinates referring to positive charges mass m x , and an 

 even index for negative charges mass m 2 . If r l = a l -\-p 1 and 

 di + cot attach to a displaced position, we can put a 2 = a±=l 

 and a 1 = a 3 = V3, ignoring the jot correction to the ratio 

 a Y : a 2 . The potential energy is given by 



U/€ 8 =D- r (l, 3) +D~ 1 (2, 4) -D-^l, 2) 

 -D- 1 (1,4)-D- 1 (3,2)«D- 1 (3,4), 



