﻿1096 Mr. R. Hargreaves on Atomic Systems 



that is p 2 = — 1 ; and subtracting forms for z 2 and £ 4 , we 

 bave p 2 — 0. Addition leads to 



(P 2 + s 3 s(9 + y/») } Ol * si) = 2 % (9 + </»)(=. + *J , 

 and 



{^ 2 +A( 9 +v / ^)}(>2+^)=A(9 + v / 3)(^i+-3); 



which give 



p * = or /»p»=-&(9-+v/3), 



that is p= +47 , 38v / — 1. 



Axial displacement therefore gives rise to true oscillations 

 for both types P x and P 2 . 



§ 28. The discovery of tangential instability prompted 

 inquiry as to the position in the system of multiple core. 

 It is convenient to use coordinates relative to the core. In 

 general, for a group comprising a mass m and other masses 

 m r , the relative kinetic energy is given by 



2T r = %m r h r 2 -(Xm r hrfl(jn Q + %m r ), . . (33) 



r 



when coordinates relative to m are used. In the present 

 case the omission of the last term involves a modification of 

 periods of the order /jl. In seeking the main term in the 

 problem of two electrons and a core + 2e, we may therefore 

 write 



2%=m 2 {k 2 + f + x' 2 + y' 2 ), 

 and pass to polar coordinates 1 + p and 6 + wt. 



Since U 2 /e 2 = ^V) 2 + i 1 6(0-0') 2 -V-V 2 , 



the equations of oscillation, taking account of ??? 2 <o 2 = 7e 2 /4, 

 are 



fe+2p P =-^(0-e'), 



P w + 2pp'=uo-ff), 



which give 



(p*-f)(p-p') = 2p{e-0'), (f + })(0-0')=-2p(p^p') 

 and 



( P *-i)(p+p')=2p(0+e'), p\e+e , ) = -2p(p+p ) . 



\. . . (34) 



