﻿Molecules 
  of 
  the 
  Elements 
  and 
  their 
  Compounds. 
  G7 
  

   Hence, 
  by 
  (1) 
  

  

  L 
  = 
  ii 
  B= 
  £(- 
  aC+a 
  ' 
  c 
  'X 
  l+ 
  ")"- 
  (38) 
  

  

  The 
  integral 
  \F 
  x 
  dt 
  between 
  the 
  limits 
  zero 
  and 
  infinity 
  can 
  

   be 
  shown 
  to 
  be 
  zero. 
  

  

  General 
  Expressions 
  for 
  Average 
  Forces. 
  

  

  The 
  above 
  equations 
  (34) 
  to 
  (37) 
  are 
  sufficiently 
  approxi- 
  

   mate 
  for 
  general 
  use, 
  as 
  they 
  include 
  the 
  fourth 
  power 
  of 
  u 
  

   in 
  the 
  development 
  of 
  (1 
  + 
  «)""*, 
  as 
  far 
  as 
  those 
  terms 
  

   are 
  concerned 
  which 
  contribute 
  anything 
  to 
  v* 
  and 
  v 
  6 
  . 
  

   v 
  s 
  is 
  not 
  included. 
  In 
  some 
  cases 
  it 
  is 
  desirable 
  to 
  include 
  

   more 
  terms 
  for 
  a 
  closer 
  approximation, 
  and 
  the 
  following 
  

   method 
  makes 
  it 
  possible 
  to 
  write 
  out 
  the 
  average 
  forces 
  

   along 
  and 
  perpendicular 
  to 
  r 
  to 
  any 
  power 
  of 
  u. 
  

  

  Let 
  T 
  = 
  asin 
  [at 
  + 
  0)-a'sin 
  (»* 
  + 
  ^ 
  r 
  )=aS.-a 
  / 
  S'. 
  (39) 
  

  

  Then, 
  from 
  (28), 
  the 
  average 
  value 
  of 
  the 
  first 
  component 
  

   resolved 
  along 
  r 
  is 
  

  

  fci- 
  rf 
  ~ 
  K^( 
  y 
  ' 
  2 
  ~ 
  ,rT 
  ) 
  [ 
  1+ 
  1 
  • 
  ?(^™*+' 
  T 
  ) 
  

  

  3 
  5 
  2V 
  \ 
  2 
  

  

  + 
  2'i's*\ 
  " 
  ) 
  

  

  + 
  i-U-?( 
  » 
  )'■•><«>> 
  

  

  Calculation 
  shows 
  that 
  t 
  llm 
  T'M 
  = 
  if 
  n 
  is 
  odd. 
  

   If 
  it 
  is 
  even, 
  we 
  find 
  the 
  following 
  : 
  — 
  

  

  where 
  g 
  = 
  a 
  2 
  + 
  a' 
  2 
  — 
  2aa' 
  cosy 
  

  

  2n 
  — 
  l 
  

   and 
  a 
  2 
  n=— 
  - 
  — 
  a 
  2n 
  _ 
  2 
  . 
  

  

  lim 
  1 
  

  

  | 
  W 
  = 
  a 
  =1, 
  

  

  J5 
  

  

  T 
  2 
  tZ^ 
  = 
  a 
  2 
  g 
  

  

  v 
  

  

  T 
  4 
  dt 
  = 
  a 
  i 
  f 
  2 
  

  

  » 
  

  

  T'dt 
  = 
  a 
  6 
  g' 
  

  

  5) 
  

  

  Vdt 
  = 
  a^ 
  

  

  55 
  

  

  T 
  2n 
  dt=a 
  2 
  n9 
  n 
  . 
  

  

  F 
  2 
  

  

  