﻿of 
  Atoms 
  and 
  Molecules. 
  861 
  

  

  By 
  help 
  of 
  this 
  expression 
  and 
  using 
  the 
  table 
  for 
  p 
  n 
  ,o—pn,m 
  

   given 
  on 
  p. 
  482 
  in 
  Part 
  II., 
  it 
  can 
  be 
  simply 
  shown 
  that 
  the 
  

   system 
  in 
  question 
  will 
  not 
  be 
  stable 
  unless 
  S"=l 
  and 
  n 
  equal 
  

   to 
  2 
  or 
  3. 
  

  

  In 
  considering 
  the 
  stability 
  o£ 
  the 
  systems 
  for 
  a 
  displace- 
  

   ment 
  of 
  the 
  nuclei 
  relative 
  to 
  each 
  other, 
  we 
  shall 
  assume 
  

   that 
  the 
  motions 
  of 
  the 
  nuclei 
  are 
  so 
  slow 
  that 
  the 
  state 
  of 
  

   motion 
  of 
  the 
  electrons 
  at 
  any 
  moment 
  will 
  not 
  differ 
  sensibly 
  

   from 
  that 
  calculated 
  on 
  the 
  assumption 
  that 
  the 
  nuclei 
  are 
  

   at 
  rest. 
  This 
  assumption 
  is 
  permissible 
  on 
  acount 
  of 
  the 
  

   great 
  mass 
  of 
  the 
  nuclei 
  compared 
  with 
  that 
  of 
  the 
  electrons, 
  

   which 
  involves 
  that 
  the 
  vibrations 
  resulting 
  from 
  a 
  displace- 
  

   ment 
  of 
  the 
  nuclei 
  are 
  very 
  slow 
  compared 
  with 
  those 
  due 
  to 
  

   a 
  displacement 
  of 
  the 
  electrons. 
  For 
  a 
  system 
  consisting 
  of 
  

   a 
  ring 
  of 
  electrons 
  and 
  two 
  nuclei 
  of 
  equal 
  charge, 
  we 
  shall 
  

   thus 
  assume 
  that 
  the 
  electrons 
  at 
  any 
  moment 
  daring 
  the 
  

   displacement 
  of 
  the 
  nuclei 
  move 
  in 
  circular 
  orbits 
  in 
  the 
  

   plane 
  of 
  symmetry 
  of 
  the 
  latter. 
  

  

  Let 
  us 
  now 
  imagine 
  that, 
  by 
  help 
  of 
  extraneous 
  forces 
  

   acting 
  on 
  the 
  nuclei, 
  we 
  slowly 
  vary 
  the 
  distance 
  between 
  

   them. 
  During 
  the 
  displacement 
  the 
  radius 
  of 
  the 
  ring 
  of 
  

   electrons 
  will 
  vary 
  in 
  consequence 
  of 
  the 
  alteration 
  of 
  

   the 
  radial 
  force 
  due 
  to 
  the 
  attraction 
  of 
  the 
  nuclei. 
  

   During 
  this 
  variation 
  the 
  angular 
  momentum 
  of 
  each 
  of 
  

   the 
  electrons 
  round 
  the 
  line 
  connecting 
  the 
  nuclei 
  will 
  

   remain 
  constant. 
  If 
  the 
  distance 
  apart 
  of 
  the 
  nuclei 
  

   increases, 
  the 
  radius 
  of 
  the 
  ring 
  will 
  obviously 
  also 
  increase 
  ; 
  

   the 
  radius, 
  however, 
  will 
  increase 
  at 
  a 
  slower 
  rate 
  than 
  the 
  

   distance 
  between 
  the 
  nuclei 
  . 
  For 
  example, 
  imagine 
  a 
  dis- 
  

   placement 
  in 
  which 
  the 
  distance 
  as 
  well 
  as 
  the 
  radius 
  are 
  

   both 
  increased 
  to 
  a 
  times 
  their 
  original 
  value. 
  In 
  the 
  new 
  

   configuration 
  the 
  radial 
  force 
  acting 
  on 
  an 
  electron 
  from 
  the 
  

  

  nuclei 
  and 
  the 
  other 
  electrons 
  is 
  —, 
  times 
  that 
  in 
  the 
  original 
  

  

  a 
  2 
  to 
  

  

  configuration. 
  From 
  the 
  constancy 
  of 
  the 
  angular 
  mo- 
  

   mentum 
  of 
  the 
  electrons 
  during 
  the 
  displacement, 
  it 
  further 
  

   follows 
  that 
  the 
  velocity 
  of 
  the 
  electrons 
  in 
  the 
  new 
  con- 
  

   figuration 
  is 
  - 
  times, 
  and 
  the 
  centrifugal 
  force 
  -s 
  times 
  that 
  

   a 
  ° 
  a. 
  6 
  

  

  in 
  the 
  original. 
  Consequently, 
  the 
  radial 
  force 
  is 
  greater 
  

   than 
  the 
  centrifugal 
  force. 
  

  

  On 
  account 
  of 
  the 
  distance 
  between 
  the 
  nuclei 
  increasing- 
  

   faster 
  than 
  the 
  radius 
  of 
  the 
  ring, 
  the 
  attraction 
  on 
  one 
  of 
  the 
  

   nuclei 
  due 
  to 
  the 
  ring 
  will 
  be 
  greater 
  than 
  the 
  repulsion 
  from 
  

   the 
  other 
  nucleus. 
  The 
  work 
  done 
  during 
  the 
  disp'acement 
  

   by 
  the 
  extraneous 
  forces 
  acting 
  on 
  the 
  nuclei 
  will 
  therefore 
  

  

  