﻿

  of 
  Atoms 
  and 
  Molecules. 
  481 
  

  

  mechanics. 
  We 
  have, 
  on 
  the 
  contrary, 
  assumed 
  that 
  the 
  

   frequency 
  of 
  the 
  radiation 
  is 
  determined 
  by 
  the 
  condition 
  

   7iv=E, 
  where 
  v 
  is 
  the 
  frequency, 
  h 
  Planck's 
  constant, 
  andE 
  

   the 
  difference 
  in 
  energy 
  corresponding 
  to 
  two 
  different 
  

   ik 
  stationary" 
  states 
  of 
  the 
  system. 
  

  

  In 
  considering 
  the 
  stability 
  of 
  a 
  ring 
  of 
  electrons 
  rotating 
  

   round 
  a 
  nucleus 
  for 
  displacements 
  of 
  the 
  electrons 
  perpen- 
  

   dicular 
  to 
  the 
  plane 
  of 
  the 
  ring, 
  imagine 
  a 
  configuration 
  

   of 
  the 
  system 
  in 
  which 
  the 
  electrons 
  are 
  displaced 
  by 
  

   •8z 
  l9 
  Sz 
  2 
  , 
  .... 
  Sz 
  n 
  respectively, 
  and 
  suppose 
  that 
  the 
  electrons, 
  

   under 
  influence 
  of 
  extraneous 
  forces, 
  rotate 
  in 
  circular 
  orbits 
  

   parallel 
  to 
  the 
  original 
  plane 
  with 
  the 
  same 
  radii 
  and 
  the 
  

   same 
  angular 
  momentum 
  round 
  the 
  axis 
  of 
  the 
  system 
  as 
  

   before. 
  The 
  kinetic 
  energy 
  is 
  unaltered 
  by 
  the 
  displacement, 
  

   •and 
  neglecting 
  powers 
  of 
  the 
  quantities 
  8z 
  h 
  .... 
  Sz 
  n 
  higher 
  

   than 
  the 
  second, 
  the 
  increase 
  of 
  the 
  potential 
  energy 
  of 
  the 
  

   system 
  is 
  given 
  by 
  

  

  J$NW^!«»eo 
  •' 
  

  

  (Sz 
  r 
  -Sz 
  s 
  y, 
  

  

  where 
  a 
  is 
  the 
  radius 
  of 
  the 
  ring, 
  Ne 
  the 
  charge 
  on 
  the 
  

   nucleus, 
  and 
  n 
  the 
  number 
  of 
  electrons. 
  According 
  to 
  the 
  

   condition 
  of 
  stability 
  in 
  § 
  1 
  the 
  system 
  is 
  stable 
  for 
  the 
  dis- 
  

   placements 
  considered, 
  if 
  the 
  above 
  expression 
  is 
  positive 
  for 
  

  

  arbitrary 
  values 
  of 
  Bz 
  x 
  , 
  8z 
  n 
  . 
  By 
  a 
  simple 
  calculation 
  it 
  

  

  can 
  be 
  shown 
  that 
  the 
  latter 
  condition 
  is 
  equivalent 
  to 
  the 
  

   condition 
  

  

  N> 
  j 
  p„ 
  >0 
  — 
  p 
  n 
  , 
  m 
  , 
  (5) 
  

  

  where 
  m 
  denotes 
  the 
  whole 
  number 
  (smaller 
  than 
  n) 
  for 
  

   which 
  

  

  p 
  n 
  k= 
  o 
  * 
  cos 
  2/c 
  — 
  cosec 
  rf 
  — 
  

   o 
  s= 
  i 
  n 
  n 
  

  

  has 
  its 
  smallest 
  value. 
  This 
  condition 
  is 
  identical 
  with 
  the 
  

   condition 
  of 
  stability 
  for 
  displacements 
  of 
  the 
  electrons 
  per- 
  

   pendicular 
  to 
  the 
  plane 
  of 
  the 
  ring, 
  deduced 
  by 
  help 
  of 
  

   ordinary 
  mechanical 
  considerations 
  *. 
  

  

  A 
  suggestive 
  illustration 
  is 
  obtained 
  by 
  imagining 
  that 
  

   the 
  displacements 
  considered 
  are 
  produced 
  by 
  the 
  effect 
  of 
  

   extraneous 
  forces 
  acting 
  on 
  the 
  electrons 
  in 
  a 
  direction 
  

   parallel 
  to 
  the 
  axis 
  of 
  the 
  ring. 
  If 
  the 
  displacements 
  are 
  

   produced 
  infinitely 
  slowly 
  the 
  motion 
  of 
  the 
  electrons 
  will 
  

   at 
  any 
  moment 
  be 
  parallel 
  to 
  the 
  original 
  plane 
  of 
  the 
  ring, 
  

   and 
  the 
  angular 
  momentum 
  of 
  each 
  of 
  the 
  electrons 
  round 
  

   * 
  Comp. 
  J. 
  W. 
  Nicholson, 
  Month. 
  Not. 
  Roy. 
  Astr. 
  Soc. 
  72. 
  p. 
  52(1912) 
  . 
  

  

  