﻿Molecules 
  of 
  the 
  Elements 
  and 
  their 
  Compounds. 
  41 
  

  

  the 
  distance 
  rapidly 
  diminishes 
  with 
  increasing 
  distance 
  to 
  a 
  

   value 
  which 
  is 
  less 
  than 
  the 
  gravitational 
  attraction 
  between 
  

   the 
  atoms. 
  Without 
  assigning 
  any 
  cause 
  for 
  the 
  existence 
  

   of 
  the 
  gravitational 
  force, 
  no 
  one 
  doubts 
  that 
  there 
  is 
  a 
  small 
  

   force 
  between 
  the 
  atoms 
  varying 
  as 
  the 
  inverse 
  square 
  of 
  the 
  

   distance. 
  Tiiis 
  force, 
  no 
  matter 
  how 
  small, 
  must 
  at 
  great 
  

   distances 
  exceed 
  any 
  force 
  varying 
  as 
  the 
  inverse 
  fourth 
  

   power, 
  and 
  at 
  small 
  distances 
  must 
  be 
  negligible 
  com- 
  

   paratively. 
  There 
  is, 
  therefore, 
  a 
  second 
  critical 
  distance 
  

   where 
  the 
  forces 
  varying 
  as 
  the 
  inverse 
  second 
  and 
  fourth 
  

   powers 
  of 
  the 
  distance 
  become 
  equal 
  to 
  each 
  other. 
  This 
  

   critical 
  distance 
  explains 
  the 
  existence 
  of 
  what 
  is 
  known 
  as 
  

   molecular 
  range, 
  a 
  distance 
  at 
  which 
  the 
  law 
  of 
  gravity 
  

   changes. 
  This 
  is 
  probably 
  many 
  times 
  greater 
  than 
  the 
  

   first 
  critical 
  distance 
  giving 
  the 
  dimensions 
  of 
  the 
  molecule. 
  

  

  Method 
  of 
  calculating 
  the 
  Forces 
  between 
  Atoms. 
  

  

  The 
  manner 
  in 
  which 
  the 
  forces 
  between 
  the 
  atoms 
  have 
  

   been 
  calculated 
  from 
  the 
  equations 
  (1) 
  to 
  (4) 
  for 
  the 
  instan- 
  

   taneous 
  force 
  between 
  two 
  moving 
  charges 
  is 
  as 
  follows. 
  

   Two 
  electrons 
  are 
  first 
  assumed 
  to 
  be 
  moving 
  each 
  at 
  a 
  

   uniform 
  rate 
  in 
  some 
  fixed 
  circular 
  orbit, 
  as 
  they 
  would 
  if 
  

   each 
  formed 
  a 
  part 
  of 
  two 
  separate 
  atoms, 
  and 
  the 
  whole 
  atom 
  

   is 
  not 
  supposed 
  to 
  move 
  from 
  one 
  fixed 
  position 
  while 
  the 
  

   electron 
  executes 
  several 
  revolutions 
  within 
  it. 
  Expressions 
  

   for 
  the 
  instantaneous 
  values 
  of 
  these 
  four 
  component 
  forces, 
  

   as 
  functions 
  of 
  the 
  time, 
  are 
  then 
  obtained. 
  As 
  each 
  depends 
  

   upon 
  the 
  instantaneous 
  distance 
  between 
  the 
  two 
  electrons, 
  

   it 
  is 
  required 
  to 
  6nd 
  this 
  distance 
  as 
  a 
  function 
  of 
  the 
  time. 
  

   When 
  the 
  distance 
  between 
  the 
  centres 
  of 
  the 
  two 
  orbits 
  is 
  so 
  

   large 
  compared 
  with 
  the 
  radii 
  o 
  f 
  ' 
  the 
  orbits 
  that 
  all 
  distances 
  

   from 
  any 
  part 
  of 
  one 
  orbit 
  to 
  the 
  other 
  are 
  sensibly 
  parallel, 
  

   the 
  expression 
  for 
  the 
  distance 
  between 
  the 
  electrons 
  is 
  a 
  

   comparatively 
  simple 
  function 
  of 
  the 
  time; 
  but, 
  when 
  the 
  

   centres 
  of 
  the 
  orbits 
  are 
  as 
  near 
  to 
  each 
  other 
  as 
  you 
  please, 
  

   and 
  their 
  axes 
  turned 
  at 
  any 
  angle, 
  and 
  the 
  rates 
  of 
  revolution 
  

   different, 
  the 
  complete 
  value 
  of 
  the 
  square 
  of 
  the 
  distance 
  is 
  

   given 
  by 
  equation 
  (11). 
  R 
  is 
  the 
  instantaneous 
  distance 
  

   between 
  the 
  electrons 
  ; 
  a 
  and 
  a 
  1 
  the 
  radii 
  of 
  the 
  orbits 
  ; 
  

   (d 
  and 
  (o' 
  the 
  angular 
  velocities 
  j 
  as, 
  ?/, 
  and 
  z 
  the 
  components 
  

   of 
  r, 
  the 
  constant 
  distance 
  between 
  the 
  centres 
  of 
  the 
  orbits, 
  

   referred 
  to 
  a 
  set 
  of 
  rectangular 
  axes 
  with 
  origin 
  at 
  the 
  centre 
  

   of 
  the 
  orbit 
  of 
  electron, 
  e. 
  The 
  ?/-axis 
  is 
  assumed 
  to 
  be 
  

   parallel 
  to 
  the 
  line 
  of 
  intersection 
  of 
  the 
  planes 
  of 
  the 
  two 
  

   orbits, 
  and 
  the 
  ar-axis 
  is 
  in 
  the 
  plane 
  of 
  the 
  orbit 
  of 
  e. 
  the 
  

  

  