﻿1094 Mr. R. Hargreaves on Atomic Systems 



and we are concerned with, the second order terms in p, #, 



viz. : 



U^=±[5(p s -p 1 )(0 i -d 2 )+ V30»*-ftj.(fli-ffij] 



+ 5(.p 1 2 + p 3 2 ) +3v / 3(p 1 +p 3 )(p 2 + p 4 )] 



■ (26) 



16 ' 48 



+ (d 4 -e 1 y+(o 3 -e 2 y+ (0 4 -0 8 ?)]. , 



To deal with oscillations of type P x this form is reduced 

 by rise of 



2 =0 ..., with the result 



TJ /__(9- V3> 2 



144 



[2( Pl + / )3) 2 + 3(^-^) 2 ] 



■WItlG)" 



l2(fi,+ Pt y+3(&- tr e l )'].. . . (27) 



When each displacement varies as e ptat , the equations of 

 ^oscillations Pi are 



(f-l) Pl -2 s/Zpd^p.+ps, 3^4-2 \%*=HK*s-*0) (28) 

 (f-l)p z -Z <s/ty0*=Pi+P*> 3/4 + 2 ^3 m - + |(4-^)'l 



leading to /> 2 = and jo 2 = — 1, the last occurring three times. 

 If p, were taken into account these roots would be unequal, 

 and we find for the type P 1? stable oscillations in the plane 

 with a period near to that of orbital revolution. 



For the group P 2 , coordinates of odd index are omitted 

 from U 2 , and 



v 2 n = ^[5p 2 2 ^5p 4 '+8p 2 p i -w 2 '-7e i 2 -4,e 2 e,]. (29) 



The equations of oscillation are then 



e 2 e 2 



m 2 p 2 co 2 p 2 ==—jn ('fy 2 + 4/> 4 ), m 2 p 2 co 2 2 = ^ {16 2 + 20 4 ) , 



16 



16 



mrf» % = - ~ (5p 4 + 4/i 2 ), m»p 2 VL 2 # 4 = Ta( 7d * + 20 *)> 



16 



16 



