﻿of 
  Atoms 
  and 
  Molecules. 
  859 
  

  

  § 
  2. 
  Configurations 
  and 
  Stability 
  of 
  the 
  Systems. 
  

  

  Let 
  us 
  consider 
  a 
  system 
  consisting 
  of 
  two 
  positive 
  nuclei 
  

   of 
  equal 
  charges 
  and 
  a 
  ring 
  of 
  electrons 
  rotating 
  round 
  the 
  

   line 
  connecting 
  them. 
  Let 
  the 
  number 
  of 
  electrons 
  in 
  the 
  

   ring 
  be 
  n, 
  the 
  charge 
  of 
  an 
  electron 
  — 
  -<?, 
  and 
  the 
  charge 
  on 
  

   each 
  nucleus 
  N<?. 
  As 
  can 
  be 
  simply 
  shown, 
  the 
  system 
  will 
  

   be 
  in 
  equilibrium 
  if 
  the 
  nuclei 
  are 
  the 
  same 
  distance 
  apart 
  

   from 
  the 
  plane 
  of 
  the 
  ring 
  and 
  if 
  the 
  ratio 
  between 
  the 
  

   diameter 
  of 
  the 
  ring 
  2a 
  and 
  the 
  distance 
  apart 
  of 
  the 
  nuclei 
  

   2b 
  is 
  given 
  by 
  

  

  b 
  = 
  

  

  <(S)*-')" 
  J 
  m 
  

  

  provided 
  thut 
  the 
  frequency 
  of 
  revolution 
  co 
  is 
  of 
  a 
  magnitude 
  

  

  such 
  that 
  for 
  each 
  of 
  the 
  electrons 
  the 
  centrifugal 
  force 
  

  

  balances 
  the 
  radial 
  force 
  due 
  to 
  the 
  attraction 
  of 
  the 
  nuclei 
  

  

  and 
  the 
  repulsion 
  of 
  the 
  other 
  electrons. 
  Denoting 
  this 
  

  

  e 
  2 
  

   force 
  by 
  -gF, 
  we 
  get 
  from 
  the 
  condition 
  of 
  the 
  universal 
  

  

  constancy 
  of 
  the 
  angular 
  momentum 
  of 
  the 
  electrons, 
  as 
  

   shown 
  in 
  Part 
  II. 
  p. 
  478, 
  

  

  a 
  =T^'2~^~ 
  l 
  and 
  co=— 
  ,-^F 
  2 
  . 
  . 
  . 
  (2) 
  

  

  h 
  

  

  The 
  total 
  energy 
  necessary 
  to 
  remove 
  all 
  the 
  charged 
  particles 
  

   to 
  infinite 
  distances 
  from 
  each 
  other 
  is 
  equal 
  to 
  the 
  total 
  

   kinetic 
  energy 
  of 
  the 
  electrons 
  and 
  is 
  given 
  by 
  

  

  W= 
  * 
  SF 
  8 
  (3) 
  

  

  IV 
  

  

  For 
  the 
  system 
  in 
  question 
  we 
  have 
  

  

  *=!W-')*-- 
  • 
  ■ 
  « 
  

  

  ■here 
  

  

  s=n-l 
  

  

  S7T 
  

  

  n 
  

  

  s 
  n 
  = 
  X 
  cosec 
  - 
  

  

  a 
  table 
  of 
  s 
  n 
  is 
  given 
  in 
  Part 
  II. 
  on 
  p. 
  482. 
  

  

  To 
  test 
  the 
  stability 
  of 
  the 
  system 
  we 
  have 
  to 
  consider 
  

   displacements 
  of 
  the 
  orbits 
  of 
  the 
  electrons 
  relative 
  to 
  the 
  

   nuclei, 
  and 
  also 
  displacements 
  of 
  the 
  latter 
  relative 
  to 
  each 
  

   other. 
  

  

  A 
  calculation 
  based 
  on 
  the 
  ordinary 
  mechanics 
  gives 
  that 
  

  

  