﻿484 
  Dr. 
  N. 
  Bohr 
  on 
  the 
  Constitution 
  

  

  rings 
  of 
  electrons 
  which 
  rotate 
  in 
  the 
  same 
  plane 
  round 
  a 
  

   nucleus 
  of 
  charge 
  N<?. 
  Let 
  the 
  radii 
  of 
  the 
  rings 
  be 
  a 
  l9 
  a 
  2 
  , 
  . 
  . 
  . 
  . 
  , 
  

   and 
  the 
  number 
  of 
  electrons 
  on 
  the 
  different 
  rings 
  %, 
  

  

  Putting 
  — 
  = 
  tan 
  2 
  (a 
  r 
  ,*) 
  5 
  we 
  g 
  e 
  f 
  £° 
  r 
  t 
  ne 
  radial 
  force 
  acting 
  

   on 
  an 
  electron 
  in 
  the 
  rth 
  ring 
  — 
  gF„ 
  where 
  

  

  i 
  ? 
  f 
  =N- 
  S 
  ,-S»,Q(«,,,); 
  

  

  the 
  summation 
  is 
  to 
  be 
  taken 
  over 
  all 
  the 
  rings 
  except 
  the 
  

  

  one 
  considered. 
  

  

  If 
  we 
  know 
  the 
  distribution 
  of 
  the 
  electrons 
  in 
  the 
  

  

  different 
  rings, 
  from 
  the 
  relation 
  (1) 
  on 
  p. 
  478, 
  we 
  can, 
  by 
  

  

  help 
  of 
  the 
  above, 
  determine 
  a 
  1} 
  a 
  2 
  , 
  . 
  . 
  . 
  . 
  The 
  calculation 
  

  

  can 
  be 
  made 
  by 
  successive 
  approximations, 
  starting 
  from 
  a 
  

  

  set 
  of 
  values 
  for 
  the 
  a's, 
  and 
  from 
  them 
  calculating 
  the 
  F's, 
  

  

  and 
  then 
  redetermining 
  the 
  a's 
  by 
  the 
  relation 
  (1) 
  which 
  

  

  F 
  a 
  

   gives 
  —^ 
  = 
  — 
  =tan 
  2 
  (a 
  r>s 
  ), 
  and 
  so 
  on. 
  

  

  -T 
  r 
  Cl 
  s 
  

  

  As 
  in 
  the 
  case 
  of 
  a 
  single 
  ring 
  it 
  is 
  supposed 
  that 
  the 
  

   systems 
  are 
  stable 
  for 
  displacements 
  of 
  the 
  electrons 
  in 
  the 
  

   plane 
  of 
  their 
  orbits. 
  In 
  a 
  calculation 
  such 
  as 
  that 
  on 
  p. 
  480, 
  

   the 
  interaction 
  of 
  the 
  rings 
  ought 
  strictly 
  to 
  be 
  taken 
  into 
  

   account. 
  This 
  interaction 
  will 
  involve 
  that 
  the 
  quantities 
  F 
  

   are 
  not 
  constant, 
  as 
  for 
  a 
  single 
  ring 
  rotating 
  round 
  a 
  nucleus, 
  

   but 
  will 
  vary 
  with 
  the 
  radii 
  of 
  the 
  rings 
  ; 
  the 
  variation 
  in 
  F, 
  

   however, 
  if 
  the 
  ratio 
  between 
  the 
  radii 
  of 
  the 
  rings 
  is 
  not 
  

   very 
  near 
  to 
  unity, 
  will 
  be 
  too 
  small 
  to 
  be 
  of 
  influence 
  on 
  

   the 
  result 
  of 
  the 
  calculation. 
  

  

  Considering 
  the 
  stability 
  of 
  the 
  systems 
  for 
  a 
  displacement 
  

  

  of 
  the 
  electrons 
  perpendicular 
  to 
  the 
  plane 
  of 
  the 
  rings, 
  it 
  is 
  

  

  necessary 
  to 
  distinguish 
  between 
  displacements 
  in 
  which 
  the 
  

  

  centres 
  of 
  gravity 
  of 
  the 
  electrons 
  in 
  the 
  single 
  rings 
  are 
  

  

  unaltered, 
  and 
  displacements 
  in 
  which 
  all 
  the 
  electrons 
  

  

  inside 
  the 
  same 
  ring 
  are 
  displaced 
  in 
  the 
  same 
  direction. 
  

  

  The 
  condition 
  of 
  stability 
  for 
  the 
  first 
  kind 
  of 
  displacements 
  

  

  is 
  given 
  by 
  the 
  condition 
  (5) 
  on 
  p. 
  481, 
  if 
  for 
  every 
  ring 
  we 
  

  

  replace 
  N 
  by 
  a 
  quantitv 
  GS> 
  determined 
  by 
  the 
  condition 
  

  

  e 
  2 
  

   that 
  — 
  ^G 
  r 
  Bz 
  is 
  equal 
  to 
  the 
  component 
  perpendicular 
  to 
  the 
  

   a 
  r 
  

  

  plane 
  of 
  the 
  ring 
  of 
  the 
  force 
  — 
  due 
  to 
  the 
  nucleus 
  and 
  the 
  

  

  electrons 
  in 
  the 
  other 
  rings 
  — 
  acting 
  on 
  one 
  of 
  the 
  electrons 
  

  

  if 
  it 
  has 
  received 
  a 
  small 
  displacement 
  Bz. 
  Using 
  the 
  same 
  

  

  notation 
  as 
  above, 
  we 
  get 
  

  

  G,. 
  = 
  N 
  — 
  Xn 
  e 
  Tl 
  (u 
  rf 
  g) 
  . 
  

  

  