546 Prof. J. W. Nicholson on the High-frequency 

 we find that the resultant force on P due to Q and R is 



e%b-2a)l(a 2 + b 2 -ab)i, 

 along PO, towards 6. The equation of steady motion of P 



is therefore 



maco { 



/ _1 \ X _1_ b-2a 



\r y/y a 2 ~~(a + b) 2 + (a 2 + b 2 - db)l ' 



where the term with -y/3 is the ordinary effect of the other 

 electrons in the same ring as P. The equation for Q is. 

 similarly 



mbco 2 /^ 1\ 1 1 a — 2b 



e 2 = V ^/3)b 2 ~ (a + b) 2+ (a 2 + b 2 -ab)V 

 If b/a = A\ we find that x is a root of the equation 



(•*-^)( 1 4) + ^Sbj I -' 1+ '>-( 1 -^ 



x = l is a root, and all others are given by 



::|iW \ \/ 3 / ^ + (l-#+O f ' 



which has no real positive roots, whatever integer is put 

 for N. The only solution is therefore b = a, so that all the 

 electrons are in the same ring. For attractive forces between 

 the electrons, on the other hand, there would be another 

 solution, even for this small number of electrons, — suggesting 

 that the essential difference between atomic problems and 

 the phenomena of Saturn's rings depends on the change 

 of .sign in the force, which is actually the reason for the 

 difference, from one point of view. 



A large number of apparently possible systems of two^ 

 rings may be shown to lead to the same conclusions, and, in 

 fact, a very general proof can be given. We may, however, 

 at this point, give only an approximate treatment of the 

 general case in which the radii are presumed to be widely 

 different, for this is the case usually quoted. Let there be- 

 n and m electrons in the inner and outer rings respectively 

 of a neutral atom, so that ]$ = n + m. Then if a is veiy 

 small in comparison with &, so that the influence of the- 

 outer ring on the inner is small, we may write 



<o*= ~(N-|S„) 

 ma 6 v 



*> 2 =^(N-»-iS„,), 



