﻿of 
  Atoms 
  and 
  Molecules. 
  

  

  485 
  

  

  If 
  all 
  the 
  electrons 
  in 
  one 
  of 
  the 
  rings 
  are 
  displaced 
  in 
  the 
  

   same 
  direction 
  by 
  help 
  of 
  extraneous 
  forces, 
  the 
  displace- 
  

   ment 
  will 
  produce 
  corresponding 
  displacements 
  of 
  the 
  

   electrons 
  in 
  the 
  other 
  rings 
  ; 
  and 
  this 
  interaction 
  will 
  be 
  of 
  

   influence 
  on 
  the 
  stability. 
  For 
  example, 
  consider 
  a 
  system 
  

   of 
  m 
  concentric 
  rings 
  rotating 
  in 
  a 
  plane 
  round 
  a 
  nucleus 
  of 
  

   charge 
  Ne, 
  and 
  let 
  us 
  assume 
  that 
  the 
  electrons 
  in 
  the 
  

   different 
  rings 
  are 
  displaced 
  perpendicular 
  to 
  the 
  plane 
  by 
  

   <^i> 
  $*2* 
  " 
  ' 
  - 
  & 
  z 
  m 
  respectively. 
  With 
  the 
  above 
  notation 
  the 
  

   increase 
  in 
  the 
  potential 
  energy 
  of 
  the 
  system 
  is 
  given 
  by 
  

  

  

  - 
  S 
  n(u 
  r 
  , 
  s 
  xsz 
  r 
  -Sz 
  s 
  y 
  

  

  The 
  condition 
  of 
  stability 
  is 
  that 
  this 
  expression 
  is 
  positive 
  

   for 
  arbitrary 
  values 
  of 
  Sz 
  1 
  , 
  . 
  . 
  . 
  . 
  hz 
  m 
  . 
  This 
  condition 
  can 
  be 
  

   worked 
  out 
  simply 
  in 
  the 
  usual 
  way. 
  It 
  is 
  not 
  of 
  sensible 
  

   influence 
  compared 
  with 
  the 
  condition 
  of 
  stability 
  for 
  the 
  

   displacements 
  considered 
  above, 
  except 
  in 
  cases 
  where 
  the 
  

   system 
  contains 
  several 
  rings 
  of 
  few 
  electrons. 
  

  

  The 
  following 
  Table, 
  containing 
  the 
  values 
  of 
  Q(«) 
  and 
  

   R(a) 
  for 
  every 
  fifth 
  degree 
  from 
  a=20° 
  to 
  « 
  = 
  70°, 
  gives 
  an 
  

   estimate 
  of 
  the 
  order 
  of 
  magnitude 
  of 
  these 
  functions 
  : 
  — 
  

  

  a. 
  

  

  tan 
  2 
  a. 
  

  

  Q(«). 
  

  

  B(«). 
  

  

  20 
  

  

  0-132 
  

  

  0001 
  

  

  0002 
  

  

  25 
  

  

  0-217 
  

  

  0005 
  

  

  0-011 
  

  

  30 
  

  

  0-333 
  

  

  0-021 
  

  

  0-048 
  

  

  35 
  

  

  0-490 
  

  

  0-080 
  

  

  0-217 
  

  

  40 
  

  

  0-704 
  

  

  0373 
  

  

  1-549 
  

  

  45 
  

  

  1000 
  

  

  

  

  50 
  

  

  1-420 
  

  

  1-708 
  

  

  4-438 
  

  

  55 
  

  

  2-040 
  

  

  1-233 
  

  

  1-839 
  

  

  60 
  

  

  3-000 
  

  

  1093 
  

  

  1-301 
  

  

  65 
  

  

  4-599 
  

  

  1-037 
  

  

  1-115 
  

  

  70 
  

  

  7-548 
  

  

  1-013 
  

  

  1-041 
  

  

  tan 
  2 
  a 
  indicates 
  the 
  ratio 
  between 
  the 
  radii 
  of 
  the 
  rings 
  

  

  i 
  tan 
  2 
  (a 
  r> 
  8 
  ) 
  = 
  — 
  | 
  . 
  The 
  values 
  of 
  Q(a) 
  show 
  that 
  unless 
  the 
  

  

  ratio 
  of 
  the 
  radii 
  of 
  the 
  rings 
  is 
  nearly 
  unity 
  the 
  effect 
  of 
  

   outer 
  rings 
  on 
  the 
  dimensions 
  of 
  inner 
  rings 
  is 
  very 
  small, 
  

   .and 
  that 
  the 
  corresponding 
  effect 
  of 
  inner 
  rings 
  on 
  outer 
  is 
  

  

  