﻿Molecules 
  of 
  the 
  Elements 
  and 
  their 
  Compoinuh. 
  59 
  

  

  the 
  second 
  to 
  e 
  '. 
  Let 
  the 
  k- 
  and 
  &'-axes 
  take 
  the 
  directions 
  

   of 
  the 
  axes 
  of 
  rotation 
  of 
  e 
  and 
  e' 
  respectively, 
  making 
  an 
  

   angle 
  « 
  with 
  each 
  other. 
  The 
  i- 
  and 
  /axes 
  will, 
  therefore, 
  

   lie 
  in 
  the 
  plane 
  of 
  the 
  orbit 
  of 
  e, 
  and 
  i'- 
  and 
  /-axes 
  in 
  a 
  

   plane 
  parallel 
  to 
  the 
  orbit 
  of 
  e 
  . 
  Since 
  the 
  planes 
  of 
  the 
  two 
  

   orbits 
  in 
  general 
  intersect 
  in 
  some 
  line, 
  this 
  direction 
  may 
  be 
  

   chosen 
  for 
  the 
  direction 
  of 
  the/ 
  and 
  /-axes, 
  and 
  the 
  prime 
  

   may 
  be 
  suppressed. 
  This 
  line 
  of 
  intersection 
  is 
  perpendicular 
  

   to 
  both 
  the 
  k- 
  and 
  A-'-axes, 
  and 
  the 
  positive 
  direction 
  along 
  

   the 
  /axis 
  may 
  be 
  defined 
  as 
  the 
  vector 
  h 
  x 
  k' 
  '. 
  The 
  angle 
  

   between 
  the 
  i- 
  and 
  i'-axes 
  is, 
  therefore, 
  a, 
  and 
  that 
  between 
  

   the 
  i'- 
  and 
  &-axes 
  is 
  the 
  complement 
  of 
  a. 
  

  

  It 
  is 
  required 
  to 
  find 
  the 
  total 
  average 
  translational 
  force 
  

   exerted 
  by 
  the 
  second 
  charge 
  e 
  upon 
  the 
  charge 
  e, 
  when 
  

   averaged 
  over 
  a 
  long 
  period 
  of 
  time 
  on 
  the 
  assumption 
  that 
  

   the 
  positions 
  of 
  the 
  orbits 
  remain 
  unchanged 
  during 
  this 
  

   time. 
  This 
  average 
  force 
  will 
  be 
  found 
  by 
  resolving 
  the 
  

   instantaneous 
  forces 
  along 
  certain 
  selected 
  directions 
  and 
  

   averaging 
  them 
  in 
  these 
  directions. 
  It 
  is 
  convenient 
  in 
  

   referring 
  to 
  an 
  atom 
  to 
  speak 
  of 
  its 
  axis, 
  equator, 
  which 
  is 
  

   the 
  plane 
  of 
  the 
  orbit, 
  and 
  a 
  meridian 
  plane, 
  and 
  circle 
  of 
  

   latitude, 
  as 
  we 
  refer 
  to 
  the 
  earth. 
  The 
  three 
  directions 
  

   chosen 
  for 
  resolving 
  all 
  forces 
  are, 
  first, 
  along 
  the 
  line 
  

   joining 
  the 
  centres 
  of 
  the 
  orbits, 
  and, 
  second, 
  perpendicular 
  

   to 
  this 
  direction 
  in 
  the 
  plane 
  of 
  the 
  meridian 
  of 
  e, 
  and, 
  third, 
  

   along 
  a 
  small 
  circle 
  of 
  latitude. 
  It 
  will 
  be 
  shown 
  that 
  the 
  

   average 
  force 
  in 
  the 
  latter 
  direction 
  is 
  always 
  zero, 
  so 
  that 
  

   there 
  are 
  only 
  two 
  directions 
  in 
  which 
  the 
  force 
  must 
  be 
  

   calculated 
  to 
  obtain 
  the 
  total 
  force. 
  

  

  Let 
  the 
  position 
  of 
  the 
  charge, 
  e, 
  in 
  its 
  orbit 
  be 
  defined 
  by 
  

   the 
  equation 
  (5), 
  and 
  of 
  e' 
  by 
  (6), 
  where 
  r 
  x 
  and 
  r 
  2 
  are 
  the 
  

   vectors 
  from 
  the 
  centres 
  of 
  their 
  orbits 
  to 
  the 
  charges 
  e 
  and 
  

   e' 
  respectively. 
  

  

  r 
  t 
  = 
  [a 
  sin 
  (&)£ 
  + 
  #)] 
  i+ 
  [a 
  cos 
  («£ 
  + 
  #)]/ 
  . 
  . 
  (5) 
  

  

  r 
  2 
  =[a'&n(m't+0 
  l 
  )]i 
  / 
  +[a'cos(G>'t 
  + 
  0')]j. 
  . 
  (6) 
  

  

  Let 
  r 
  denote 
  the 
  constant 
  vector 
  from 
  the 
  centre 
  of 
  the 
  

   orbit 
  of 
  e 
  to 
  that 
  of 
  e\ 
  having 
  the 
  components 
  »r, 
  y, 
  and 
  s 
  

   referred 
  to 
  the 
  i-,j-, 
  and 
  &-axes, 
  whence 
  

  

  r 
  = 
  ,vi 
  + 
  ?/j 
  + 
  zk 
  (7) 
  

  

  The 
  vector 
  R, 
  from 
  the 
  charge 
  e 
  to 
  the 
  charge 
  e', 
  is, 
  

   therefore, 
  

  

  R=-r 
  1 
  + 
  r-hr 
  2 
  (8) 
  

  

  