﻿860 
  Dr. 
  N. 
  Bohr 
  on 
  the 
  Constitution 
  

  

  the 
  systems 
  are 
  unstable 
  for 
  displacements 
  of 
  the 
  electrons 
  in 
  

   the 
  plane 
  of 
  the 
  ring. 
  As 
  for 
  the 
  systems 
  considered 
  in 
  

   Part 
  II., 
  we 
  shall, 
  however, 
  assume 
  that 
  the 
  ordinary 
  

   principles 
  of 
  mechaDics 
  cannot 
  be 
  used 
  in 
  discussing 
  the 
  

   problem 
  in 
  question, 
  and 
  that 
  the 
  stability 
  of 
  the 
  systems 
  for 
  

   the 
  displacements 
  considered 
  is 
  secured 
  through 
  the 
  in- 
  

   troduction 
  of 
  the 
  hypothesis 
  of 
  the 
  universal 
  constancy 
  of 
  

   the 
  angular 
  momentum 
  of 
  the 
  electrons. 
  This 
  assumption 
  is 
  

   included 
  in 
  the 
  condition 
  of 
  stability 
  stated 
  in 
  § 
  1. 
  Jt 
  should 
  

   be 
  noticed 
  that 
  in 
  Part 
  II. 
  the 
  quantity 
  F 
  was 
  taken 
  as 
  a 
  

   constant, 
  while 
  for 
  the 
  systems 
  considered 
  here, 
  F, 
  for 
  fixed 
  

   positions 
  of 
  the 
  nuclei, 
  varies 
  with 
  the 
  radius 
  of 
  the 
  ring. 
  A 
  

   simple 
  calculation, 
  however, 
  similar 
  to 
  that 
  given 
  in 
  Part 
  II. 
  

   on 
  p. 
  480, 
  shows 
  that 
  the 
  increase 
  in 
  the 
  total 
  energy 
  of 
  the 
  

   system 
  for 
  a 
  variation 
  of 
  the 
  radius 
  of 
  the 
  ring 
  from 
  a 
  to 
  

   a 
  + 
  ha, 
  neglecting 
  powers 
  of 
  ha 
  greater 
  than 
  the 
  second, 
  is 
  

   given 
  by 
  

  

  a 
  ( 
  P 
  + 
  T 
  )= 
  T(i 
  + 
  ^)@\ 
  

  

  where 
  T 
  is 
  the 
  total 
  kinetic 
  energy 
  and 
  P 
  the 
  potential 
  energy 
  

   of 
  the 
  system. 
  Since 
  for 
  fixed 
  positions 
  of 
  the 
  nuclei 
  F 
  

   increases 
  for 
  increasing 
  a 
  (F 
  = 
  for 
  a~0 
  ; 
  F 
  = 
  2N 
  — 
  .<? 
  n 
  for 
  

   a 
  = 
  cc), 
  the 
  term 
  dependent 
  on 
  the 
  variation 
  of 
  F 
  will 
  be 
  

   positive, 
  and 
  the 
  system 
  will 
  consequently 
  be 
  stable 
  for 
  the 
  

   displacement 
  in 
  question. 
  

  

  From 
  considerations 
  exactly 
  corresponding 
  to 
  those 
  given 
  

   in 
  Part 
  II. 
  on 
  p. 
  481, 
  we 
  get 
  for 
  the 
  conditioirpf 
  stability 
  for 
  

   displacements 
  of 
  the 
  electrons 
  perpendicular 
  to 
  the 
  plane 
  of 
  

   the 
  ring 
  

  

  Q>Pn,o— 
  Pn,m, 
  (5) 
  

  

  where 
  p 
  n 
  ,o—pn,m 
  has 
  the 
  same 
  signification 
  as 
  in 
  Part 
  II., 
  

  

  and 
  — 
  3 
  G8z 
  denotes 
  the 
  component, 
  perpendicular 
  to 
  the 
  

  

  plane 
  of 
  the 
  ring, 
  of 
  the 
  force 
  due 
  to 
  the 
  nuclei, 
  which 
  acts 
  

   upon 
  one 
  of 
  the 
  electrons 
  in 
  the 
  ring 
  when 
  it 
  has 
  suffered 
  a 
  

   small 
  displacement 
  8z 
  perpendicular 
  to 
  the 
  plane 
  of 
  the 
  

   ring. 
  As 
  lor 
  the 
  systems 
  considered 
  in 
  Part 
  II., 
  the 
  dis- 
  

   placements 
  can 
  be 
  imagined 
  to 
  be 
  produced 
  by 
  the 
  effect 
  of 
  

   extraneous 
  forces 
  acting 
  upon 
  the 
  electrons 
  in 
  direction 
  

   parallel 
  to 
  the 
  axis 
  of 
  the 
  system. 
  

  

  For 
  a 
  system 
  of 
  two 
  nuclei 
  each 
  of 
  charge 
  Ne 
  and 
  with 
  a 
  

   ring 
  of 
  n 
  electrons, 
  we 
  find 
  

  

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