﻿Molecules 
  of 
  the 
  Elements 
  and 
  their 
  Compounds. 
  71 
  

  

  The 
  Force 
  between 
  Atoms. 
  

  

  The 
  preceding 
  calculations 
  apply 
  to 
  the 
  case 
  of: 
  two 
  moving 
  

   electrical 
  charges. 
  It 
  is 
  now 
  required 
  to 
  sum 
  the 
  forces 
  

   over 
  each 
  individual 
  part 
  of 
  two 
  atoms. 
  The 
  atoms 
  are 
  

   supposed 
  to 
  be 
  constituted 
  as 
  before 
  described 
  of 
  a 
  sphere 
  

   of 
  positive 
  electricity 
  of 
  uniform 
  density 
  within 
  which 
  the 
  

   electrons 
  lie 
  all 
  in 
  one 
  plane, 
  but 
  in 
  different 
  concentric 
  

   rings 
  in 
  that 
  plane, 
  each 
  describing 
  a 
  circular 
  orbit 
  with 
  the 
  

   same 
  angular 
  velocity. 
  In 
  the 
  case 
  where 
  the 
  distance 
  

   between 
  the 
  atoms 
  is 
  small, 
  so 
  that 
  they 
  come 
  to 
  stable 
  

   equilibrium 
  positions 
  relative 
  to 
  each 
  other, 
  it 
  will 
  be 
  

   considered 
  that 
  their 
  axes 
  of 
  revolution 
  are 
  parallel 
  to 
  each 
  

   other. 
  

  

  The 
  positive 
  sphere 
  may 
  be 
  considered 
  as 
  a 
  stationary 
  

   charge, 
  and 
  the 
  equations 
  (42) 
  and 
  (44) 
  give 
  the 
  force 
  

   between 
  a 
  moving 
  and 
  a 
  stationary 
  charge 
  if 
  thn 
  radius 
  a 
  or 
  

   a' 
  of 
  one 
  of 
  the 
  orbits 
  is 
  made 
  equal 
  to 
  zero. 
  There 
  is 
  then 
  

   no 
  phase 
  angle 
  7, 
  and 
  the 
  correct 
  value 
  of 
  the 
  force 
  is 
  

   obtained 
  if 
  all 
  terms 
  containing 
  7 
  are 
  omitted, 
  and 
  the 
  proper 
  

   sign 
  of 
  the 
  charge 
  used. 
  

  

  The 
  process 
  of 
  writing 
  down 
  the 
  terms 
  that 
  are 
  given 
  by 
  

   equations 
  (42) 
  and 
  (44) 
  for 
  the 
  forces 
  along 
  r, 
  and 
  perpen- 
  

   dicular 
  to 
  r 
  for 
  each 
  individual 
  electron 
  and 
  the 
  positive 
  

   spheres 
  in 
  the 
  atoms, 
  and 
  then 
  expanding 
  them 
  in 
  powers 
  of 
  

   the 
  distance, 
  v, 
  by 
  the 
  binomial 
  theorem, 
  is 
  rather 
  long, 
  but 
  

   it 
  is 
  straightforward. 
  Inasmuch 
  as 
  general 
  equations 
  have 
  

   been 
  obtained, 
  which 
  apply 
  to 
  any 
  atom 
  including 
  all 
  terms 
  

   in 
  the 
  expansion 
  up 
  to 
  v~ 
  6 
  inclusive, 
  but 
  not 
  including 
  v~ 
  8 
  f 
  

   an 
  example 
  of 
  the 
  process 
  is 
  omitted, 
  and 
  the 
  particular 
  

   solutions 
  obtained 
  from 
  the 
  general 
  equation. 
  

  

  If 
  m 
  denotes 
  the 
  radius 
  of 
  the 
  orbit 
  of 
  one 
  electron, 
  e, 
  in 
  

   the 
  atom 
  A, 
  referred 
  to 
  the 
  radius 
  of 
  some 
  selected 
  orbit, 
  a*, 
  

   as 
  a 
  unit, 
  so 
  that 
  the 
  radius 
  in 
  centimetres 
  is 
  a 
  = 
  ma*, 
  and 
  

   if 
  m 
  denotes 
  the 
  radius 
  of 
  the 
  orbit 
  of 
  e' 
  in 
  the 
  atom 
  A' 
  in 
  a 
  

   similar 
  manner, 
  then 
  it 
  has 
  been 
  shown 
  that 
  the 
  force 
  which 
  

   the 
  whole 
  atom 
  A 
  7 
  exerts 
  upon 
  the 
  atom 
  A 
  resolved 
  along 
  

   r 
  is 
  

  

  F 
  ==4^-4i(8-12cos 
  2 
  X)y3V-40 
  + 
  200cos 
  2 
  \~^cos 
  4 
  \"|, 
  (45) 
  

  

  along 
  x 
  ±v 
  V 
  a 
  * 
  \_ 
  J 
  

  

  and 
  the 
  force 
  resolved 
  in 
  the 
  meridian 
  plane 
  perpendicular 
  

   to 
  r 
  is 
  

  

  F 
  = 
  - 
  J 
  ^ 
  2 
  ' 
  4-2 
  i^/SV 
  - 
  40 
  + 
  70 
  cos 
  2 
  \\ 
  sin 
  2\, 
  . 
  (46) 
  

  

  