﻿480 
  Dr. 
  N. 
  Bohr 
  on 
  the 
  Constitution 
  

  

  not 
  small 
  compared 
  with 
  the 
  velocity 
  of 
  light, 
  we 
  get 
  the 
  

   same 
  expression 
  for 
  v 
  as 
  that 
  given 
  by 
  (1), 
  while 
  the 
  quantity 
  

  

  m 
  in 
  the 
  expressions 
  for 
  a 
  and 
  co 
  is 
  replaced 
  by 
  ~jj^ 
  2/^2 
  \' 
  

  

  and 
  in 
  the 
  expression 
  for 
  W 
  by 
  v 
  ^ 
  ~ 
  v 
  ' 
  c 
  ' 
  

  

  As 
  stated 
  in 
  Part 
  I., 
  a 
  calculation 
  based 
  on 
  the 
  ordinary 
  

   mechanics 
  gives 
  the 
  result, 
  that 
  a 
  ring 
  of 
  electrons 
  rotating 
  

   round 
  a 
  positive 
  nucleus 
  in 
  general 
  is 
  unstable 
  for 
  displace- 
  

   ments 
  of 
  the 
  electrons 
  in 
  the 
  plane 
  of 
  the 
  ring. 
  In 
  order 
  to 
  

   escape 
  from 
  this 
  difficulty, 
  we 
  have 
  assumed 
  that 
  the 
  ordinary 
  

   principles 
  of 
  mechanics 
  cannot 
  be 
  used 
  in 
  the 
  discussion 
  of 
  

   the 
  problem 
  in 
  question, 
  any 
  more 
  than 
  in 
  the 
  discussion 
  of 
  

   the 
  connected 
  problem 
  of 
  the 
  mechanism 
  of 
  binding 
  of 
  elec- 
  

   trons. 
  We 
  have 
  also 
  assumed 
  that 
  the 
  stability 
  for 
  such 
  

   displacements 
  is 
  secured 
  through 
  the 
  introduction 
  of 
  the 
  

   hypothesis 
  of 
  the 
  universal 
  constancy 
  of 
  the 
  angular 
  momen- 
  

   tum 
  of 
  the 
  electrons. 
  

  

  As 
  is 
  easily 
  shown, 
  the 
  latter 
  assumption 
  is 
  included 
  in 
  the 
  

   condition 
  of 
  stability 
  in 
  § 
  1. 
  Consider 
  a 
  ring 
  of 
  electrons 
  

   rotating 
  round 
  a 
  nucleus, 
  and 
  assume 
  that 
  the 
  system 
  is 
  in 
  

   dynamical 
  equilibrium 
  and 
  that 
  the 
  radius 
  of 
  the 
  ring 
  is 
  a 
  ,. 
  

   the 
  velocity 
  of 
  the 
  electrons 
  v 
  , 
  the 
  total 
  kinetic 
  energy 
  T 
  „ 
  

   and 
  the 
  potential 
  energy 
  P 
  . 
  As 
  shown 
  in 
  Part 
  I. 
  (p. 
  21) 
  

   we 
  have 
  P 
  = 
  — 
  2T 
  . 
  Next 
  consider 
  a 
  configuration 
  of 
  the 
  

   system 
  in 
  which 
  the 
  electrons, 
  under 
  influence 
  of 
  extraneous 
  

   forces, 
  rotate 
  with 
  the 
  same 
  angular 
  momentum 
  round 
  the 
  

   nucleus 
  in 
  a 
  ring 
  of 
  radius 
  a 
  = 
  aa 
  . 
  In 
  this 
  case 
  we 
  have 
  

  

  P=-P 
  , 
  and 
  on 
  account 
  of 
  the 
  uniformity 
  of 
  the 
  angular 
  

  

  a 
  1 
  1 
  

  

  momentum 
  v= 
  - 
  v 
  and 
  T=— 
  9 
  T 
  . 
  Using 
  the 
  relation 
  

  

  a 
  or 
  ° 
  

  

  P 
  =-2T„,weget 
  

  

  P 
  + 
  T=ip„+ 
  AT 
  = 
  P 
  + 
  T 
  + 
  T 
  (l--Y. 
  

  

  a. 
  ar 
  \ 
  a/ 
  

  

  We 
  see 
  that 
  the 
  total 
  energy 
  of 
  the 
  new 
  configuration 
  is 
  

   greater 
  than 
  in 
  the 
  original. 
  According 
  to 
  the 
  condition 
  of 
  

   stability 
  in 
  § 
  1 
  the 
  system 
  is 
  consequently 
  stable 
  for 
  the 
  

   displacement 
  considered. 
  In 
  this 
  connexion, 
  it 
  may 
  be 
  re- 
  

   marked 
  that 
  in 
  Part 
  I. 
  we 
  have 
  assumed 
  that 
  the 
  frequency 
  

   of 
  radiation 
  emitted 
  or 
  absorbed 
  by 
  the 
  systems 
  cannot 
  be 
  

   determined 
  from 
  the 
  frequencies 
  of 
  vibration 
  of 
  the 
  electrons 
  

   in 
  the 
  plane 
  of 
  the 
  orbits, 
  calculated 
  by 
  help 
  of 
  the 
  ordinary 
  

  

  

  