﻿Molecules 
  of 
  the 
  Elements 
  and 
  their 
  Compounds. 
  71* 
  

  

  If 
  F 
  denotes 
  the 
  sum 
  of 
  the 
  forces 
  due 
  to 
  the 
  ring 
  of 
  

   electrons, 
  the 
  central 
  electrons, 
  and 
  the 
  positive 
  sphere 
  upon 
  

   the 
  one 
  electron 
  at 
  A 
  in 
  the 
  outside 
  ring, 
  we 
  have 
  

  

  F=F'+F"+F'"+F 
  4 
  +F 
  5 
  +F 
  6 
  , 
  

  

  or 
  

  

  F 
  = 
  _ 
  f 
  ,, 
  S+ 
  ^X(2P-S)-^-, 
  + 
  ££. 
  

   4ka 
  1 
  - 
  ±ai- 
  v 
  J 
  Ka 
  L 
  " 
  Kb 
  6 
  

  

  Since 
  A 
  is 
  moving 
  in 
  a 
  circular 
  orbit, 
  its 
  acceleration 
  

   is 
  q*/ai, 
  and 
  if 
  m 
  denotes 
  its 
  mass, 
  we 
  have 
  

  

  F 
  = 
  mqfjai 
  = 
  ra^o) 
  2 
  . 
  

  

  Equating 
  these 
  forces 
  and 
  substituting 
  for 
  fi 
  its 
  equivalent 
  

   1/Kc 
  2 
  , 
  c 
  being 
  the 
  velocity 
  of 
  light, 
  and 
  multiplying 
  each 
  

   term 
  by 
  Ka 
  L 
  2 
  je 
  2 
  , 
  we 
  have 
  

  

  8 
  - 
  2 
  ~ 
  2 
  — 
  3 
  

  

  * 
  l 
  + 
  ^ 
  {2 
  F-S)-u 
  + 
  ^f 
  = 
  ^^ 
  . 
  (71) 
  

  

  -T 
  + 
  

  

  Another 
  assumption, 
  which 
  is 
  introduced 
  from 
  the 
  ring 
  

   theory, 
  is 
  that 
  the 
  volume 
  of 
  the 
  positive 
  sphere 
  is 
  propor- 
  

   tional 
  to 
  the 
  total 
  number 
  of 
  electrons 
  contained 
  therein. 
  

   If 
  b 
  and 
  6' 
  are 
  the 
  radii 
  of 
  two 
  different 
  kinds 
  of 
  atoms 
  

   containing 
  p 
  and 
  p' 
  electrons 
  respectively, 
  we 
  have 
  on 
  this 
  

   assumption 
  (ft/6') 
  3 
  = 
  p 
  'p 
  . 
  In 
  the 
  atoms 
  containing 
  two 
  

   electrons, 
  the 
  radius 
  is 
  6', 
  and 
  p' 
  = 
  2, 
  hence 
  

  

  >>* 
  = 
  p' 
  3 
  , 
  (72) 
  

  

  Substituting 
  this 
  value 
  of 
  b 
  in 
  (71), 
  we 
  find 
  

  

  / 
  1 
  mK&) 
  2 
  \ 
  

  

  2P-S«V 
  S 
  u 
  

  

  1 
  + 
  -8~ 
  c 
  2 
  " 
  = 
  8+2- 
  " 
  ™ 
  

  

  When 
  — 
  =/3 
  is 
  small, 
  the 
  radius 
  referred 
  to 
  the 
  positive 
  

  

  sphere 
  is 
  given 
  by 
  

  

  S 
  ii 
  

  

  a 
  '- 
  ,= 
  * 
  + 
  * 
  (•*) 
  

  

  or, 
  restoring 
  // 
  from 
  (72), 
  

  

  (!) 
  8= 
  jKi 
  + 
  ") 
  n: 
  '> 
  

  

  