﻿Molecules 
  of 
  the 
  Elements 
  and 
  their 
  Compounds. 
  45 
  

  

  Generally 
  the 
  sums 
  of 
  the 
  odd 
  powers 
  o£ 
  the 
  cosines 
  are 
  

   zero, 
  and 
  the 
  sums 
  of 
  the 
  even 
  powers 
  are 
  constant, 
  inde- 
  

   pendent 
  of 
  the 
  initial 
  phase-angle, 
  7, 
  except 
  in 
  the 
  very 
  

   special 
  cases 
  where 
  the 
  number 
  of 
  electrons 
  in 
  the 
  ring 
  

   is 
  small. 
  In 
  certain 
  atoms 
  this 
  number 
  is 
  small, 
  and 
  the 
  

   resulting 
  force 
  depends 
  to 
  a 
  large 
  extent 
  upon 
  the 
  relative 
  

   phase-angles 
  ; 
  but 
  in 
  the 
  majority 
  c£ 
  them 
  the 
  equilibrium 
  

   position 
  is 
  independent 
  of 
  this 
  phase-angle 
  to 
  a 
  very 
  close 
  

   degree 
  of 
  approximation. 
  To 
  illustrate 
  : 
  when 
  there 
  is 
  a 
  

   ring 
  containing 
  but 
  two 
  electrons 
  in 
  each 
  atom, 
  the 
  force 
  

   involves 
  the 
  phase-angle 
  7 
  in 
  those 
  terms 
  containing 
  the 
  

   inverse 
  sixth 
  power 
  of 
  the 
  distance, 
  #~ 
  6 
  , 
  and 
  with 
  a 
  ring 
  of 
  

   three 
  electrons 
  terms 
  do 
  not 
  appear 
  until 
  v~ 
  8 
  . 
  For 
  most 
  

   purposes 
  the 
  approximation 
  is 
  sufficiently 
  close 
  for 
  a 
  deter- 
  

   mination 
  of 
  the 
  equilibrium 
  position 
  if 
  we 
  neglect 
  all 
  terms 
  

   containing 
  v~ 
  s 
  and 
  higher 
  powers; 
  so 
  that 
  it 
  happens 
  that 
  

   in 
  all 
  the 
  cases, 
  except 
  when 
  there 
  is 
  a 
  ring 
  of 
  two 
  electrons 
  

   in 
  each 
  atom, 
  the 
  equilibrium 
  position 
  is 
  approximately 
  

   independent 
  of 
  the 
  relative 
  phase-angles 
  of 
  the 
  different 
  

   electrons. 
  

  

  Equations 
  (45) 
  and 
  (46) 
  give 
  the 
  complete 
  values 
  of 
  the 
  

   forces 
  which 
  the 
  atom 
  A' 
  exerts 
  upon 
  A 
  including 
  the 
  

   inverse 
  sixth 
  power 
  of 
  i\ 
  except 
  for 
  certain 
  terms 
  con- 
  

   taining 
  ft 
  which 
  are 
  negligible 
  in 
  comparison 
  with 
  the 
  other 
  

   terms. 
  These 
  equations 
  may 
  safely 
  be 
  used 
  £01* 
  atoms, 
  except 
  

   in 
  the 
  case 
  where 
  each 
  atom 
  has 
  one 
  ring 
  with 
  two 
  electrons. 
  

   It 
  may 
  be 
  used 
  if 
  one 
  atom 
  has 
  such 
  a 
  ring 
  and 
  the 
  other 
  

   has 
  not. 
  

  

  Table 
  III. 
  gives 
  the 
  values 
  of 
  the 
  ratios 
  m 
  l5 
  w 
  2 
  , 
  of 
  the 
  radii 
  

   of 
  the 
  outside 
  and 
  second 
  rings 
  of 
  electrons 
  in 
  several 
  selected 
  

   atoms, 
  referred 
  to 
  the 
  radius 
  of 
  the 
  single 
  ring 
  atom 
  having 
  

   but 
  three 
  electrons; 
  also 
  the 
  values 
  of 
  k 
  2 
  and 
  & 
  4 
  (48) 
  and 
  

  

  Table 
  III. 
  

  

  

  m 
  v 
  

  

  m 
  2 
  . 
  

  

  k 
  2 
  . 
  

  

  K 
  

  

  hIK 
  

  

  2,0 
  

  

  0-7566 
  

  

  0000 
  

  

  1144 
  

  

  0655 
  

  

  0-573 
  

  

  3,0 
  

  

  roooo 
  

  

  o-ooo 
  

  

  3-000 
  

  

  3000 
  

  

  1-000 
  

  

  4.1 
  

  

  T.n05 
  

  

  0000 
  

  

  9-064 
  

  

  2048 
  

  

  2-26 
  

  

  6,1 
  

  

  1-700 
  

  

  0000 
  

  

  1734 
  

  

  50-10 
  

  

  2-89 
  

  

  8,1 
  

  

  1-875 
  

  

  0000 
  

  

  28-20 
  

  

  98-80 
  

  

  3-50 
  

  

  8,2 
  

  

  2-03 
  

  

  0-500 
  

  

  33-46 
  

  

  135-63 
  

  

  4-05 
  

  

  8.3 
  

  

  2-105 
  

  

  0-689 
  

  

  39-99 
  

  

  186-52 
  

  

  466 
  

  

  9.3 
  

  

  2-255 
  

  

  0-666 
  

  

  47-05 
  

  

  232-79 
  

  

  4-95 
  

  

  9,4 
  

  

  2-385 
  

  

  0-832 
  

  

  5396 
  

  

  29316 
  

  

  5 
  44 
  

  

  10,5 
  

  

  2*515 
  

  

  0971 
  

  

  6949 
  

  

  42395 
  

  

  6-10 
  

  

  