﻿1082 Mr. R. Hargreaves on Atomic Systems 



central attraction on m 2 of amount eV 2 /a 2 2 , it will correspond 

 to using- <r 1 and o- 2 in place of the units occurring in the 

 equations (6 b). From (14 a) and (V6a,b) we find 



8x = 2a£<7 2 /N and SN = - a x + xa 2 . . (21a) 



The correction for a positive centre is then got by writing 

 <r 1 = o- 2 =l, and gives 



o> = 2*f/N and SN "= f . . . . (21 6j 



The correction for yu,, that is to take account of the ratio 

 m 2 : m L instead of assuming it indefinitely small, is got by 

 writing <j 2 = — yu,N#~ 3 , and is 



&p=-2/if«-», SN=-/*fNa;- 2 . . (21c) 



§ 15. It»is proposed to deal briefly with the potential of 

 charges distributed at equal intervals on a circle,, with a view 

 to showing the mean effect and the main fluctuation in the 

 force on a satellite. In the plane of a single ring radius a 

 we are concerned with a potential 



U/e = 2 {r 2 + a 2 -2ra cos {<j> + 2pTr/ii\}- 1 ' 2 , 



p = 



where $ is the angle between the radius r of a satellite and 

 that to the nearest unit in the ring. The Legendre co- 

 efficients must be expanded in cosines of multiples of the 

 angle <f)-t 2p7rjn, and note taken of the fact that 



»-l 



2 cos m (<f> + 2p7r/n) 



p=0 



vanishes unless m is a multiple of n, in which case it is 

 n cos m<f). The mean value of the attraction depends on the 

 term independent of <£, which is 



ov ] (22) 



rl /l\ 2 r 2 , /1.3XV , 1 • ^ v (r\ 



according as r or a is the greater. K is the first complete 

 elliptic integral. Thus for the motion of each of s satellites 

 an approximate value of the effect of the double ring is 



