﻿of 
  Atoms 
  and 
  Molecules. 
  483 
  

  

  near 
  to 
  unity, 
  the 
  conditions 
  of 
  stability 
  obtained 
  on 
  this 
  

   assumption 
  may 
  also 
  be 
  considered 
  as 
  sufficient. 
  

  

  We 
  have 
  assumed 
  in 
  § 
  1 
  that 
  the 
  electrons 
  in 
  the 
  atoms 
  

   rotate 
  in 
  coaxial 
  rings. 
  The 
  calculation 
  indicates 
  that 
  only 
  

   in 
  the 
  case 
  o£ 
  systems 
  containing 
  a 
  great 
  number 
  of 
  electrons 
  

   will 
  the 
  planes 
  of 
  the 
  rings 
  separate 
  ; 
  in 
  the 
  case 
  of 
  systems 
  

   containing 
  a 
  moderate 
  number 
  of 
  electrons, 
  all 
  the 
  rings 
  will 
  

   be 
  situated 
  in 
  a 
  single 
  plane 
  through 
  the 
  nucleus. 
  For 
  the 
  

   sake 
  of 
  brevity, 
  we 
  shall 
  therefore 
  here 
  only 
  consider 
  the 
  

   latter 
  case. 
  

  

  Let 
  us 
  consider 
  an 
  electric 
  charge 
  E 
  uniformly 
  distributed 
  

   along 
  the 
  circumference 
  of 
  a 
  circle 
  of 
  radius 
  a. 
  

  

  At 
  a 
  point 
  distant 
  z 
  from 
  the 
  plane 
  of 
  the 
  ring, 
  and 
  at 
  a 
  

   distance 
  r 
  from 
  the 
  axis 
  of 
  the 
  ring, 
  the 
  electrostatic 
  potential 
  

   is 
  given 
  by 
  

  

  u— 
  iaf" 
  * 
  

  

  l 
  V 
  {a? 
  + 
  r 
  i 
  + 
  z 
  2 
  -2ar 
  cos 
  S) 
  

  

  Putting 
  in 
  this 
  expression 
  ^=0 
  and 
  ~=tan 
  2 
  a, 
  and 
  using 
  

   the 
  notation 
  a 
  

  

  (*2 
  

   . 
  o 
  V 
  

  

  d$ 
  

  

  ••'(1 
  -sin 
  2 
  a 
  cos 
  2 
  $)' 
  

  

  we 
  get 
  for 
  the 
  radial 
  force 
  exerted 
  on 
  an 
  electron 
  in 
  a 
  point 
  

   in 
  the 
  plane 
  of 
  the 
  ring 
  

  

  where 
  

  

  Q(a)= 
  -sin 
  4 
  a(K(2a)-cotaK'(2a)). 
  

  

  The 
  corresponding 
  force 
  perpendicular 
  to 
  the 
  plane 
  of 
  the 
  

   ring 
  at 
  a 
  distance 
  r 
  from 
  the 
  centre 
  of 
  the 
  ring 
  and 
  at 
  a 
  

   small 
  distance 
  Bz 
  from 
  its 
  plane 
  is 
  given 
  by 
  

  

  ^t=-^ 
  r 
  (*)> 
  

  

  a 
  

  

  where 
  

  

  E(a) 
  = 
  -sin 
  6 
  «(K(2a) 
  + 
  tan 
  (2«)K'(2*)) 
  ; 
  

  

  A 
  short 
  table 
  of 
  the 
  functions 
  Q(a) 
  and 
  R(a) 
  is 
  given 
  

   on 
  p. 
  485. 
  

  

  Next 
  consider 
  a 
  system 
  consisting 
  of 
  a 
  number 
  of 
  concentric 
  

  

  