﻿60 
  Dr. 
  A. 
  0. 
  Crehoro 
  on 
  the 
  Formation 
  of 
  the 
  

  

  and, 
  by 
  addition, 
  we 
  find 
  

  

  IL 
  = 
  Ai-rBj 
  + 
  Ck 
  + 
  I)i 
  , 
  (9) 
  

  

  where 
  A 
  = 
  # 
  — 
  a 
  sin 
  (cot 
  + 
  6), 
  

  

  B=y 
  — 
  acos(cot 
  + 
  0) 
  + 
  a' 
  cos 
  (<o't 
  + 
  0'), 
  

  

  D=a'sin(a>'* 
  + 
  0'). 
  

  

  To 
  obtain 
  the 
  magnitude 
  of 
  R, 
  namely 
  R, 
  take 
  the 
  direct 
  

   product 
  R 
  . 
  R 
  from 
  (9), 
  and 
  find 
  

  

  R 
  2 
  = 
  A 
  2 
  + 
  B 
  2 
  + 
  C 
  2 
  + 
  D 
  2 
  + 
  2ADcos«4-2eDsin«. 
  (10) 
  

  

  Upon 
  substituting 
  the 
  values 
  of 
  A, 
  B, 
  &c, 
  in 
  (10), 
  we 
  obtain 
  

   the 
  value 
  of 
  R 
  2 
  as 
  a 
  function 
  of 
  the 
  time, 
  which 
  is 
  true 
  for 
  

   any 
  distance, 
  r, 
  between 
  the 
  centres 
  of 
  orbits, 
  and 
  for 
  any 
  

   inclination, 
  a, 
  of 
  the 
  axes 
  of 
  rotation. 
  

  

  R 
  2 
  = 
  s%l+u). 
  ...... 
  (11) 
  

  

  2 
  r 
  

   it 
  — 
  ^ 
  — 
  ax 
  sin 
  (oat 
  + 
  6)— 
  ay 
  cos 
  (<ot 
  + 
  6) 
  + 
  a'z 
  sin 
  a 
  sin 
  (w 
  r 
  t 
  + 
  0') 
  

  

  + 
  a! 
  y 
  cos 
  (j&'t 
  -f 
  0') 
  + 
  a'x 
  cos 
  a 
  sin 
  (to 
  f 
  't 
  + 
  0') 
  

  

  — 
  aa' 
  cos 
  a 
  sin 
  (oot 
  + 
  0) 
  sin 
  (a>'£ 
  + 
  0') 
  — 
  aa' 
  cos 
  (cot 
  + 
  0) 
  cos 
  (©'£ 
  + 
  0') 
  

  

  • 
  • 
  • 
  (12) 
  

   where 
  s 
  2 
  = 
  x 
  2 
  +y 
  2 
  -\-z 
  2 
  + 
  a 
  2 
  + 
  a' 
  2 
  = 
  r 
  2 
  + 
  a 
  2 
  + 
  a' 
  2 
  , 
  a 
  constant 
  . 
  . 
  . 
  (13) 
  

  

  The 
  value 
  of 
  R 
  2 
  is 
  put 
  in 
  this 
  form 
  for 
  convenience 
  in 
  

   expansion, 
  as 
  we 
  shall 
  require 
  R~ 
  3 
  in 
  obtaining 
  the 
  first 
  and 
  

   second 
  component 
  forces. 
  The 
  first 
  component 
  force 
  (1) 
  is 
  

   the 
  electrostatic 
  repulsion 
  of 
  the 
  charge 
  e 
  upon 
  that 
  of 
  e,> 
  

   and 
  it 
  acts 
  in 
  the 
  direction 
  e'e, 
  or 
  — 
  R. 
  To 
  resolve 
  this 
  

   force 
  along 
  the 
  direction 
  O'O, 
  or 
  — 
  r, 
  the 
  line 
  joining 
  the 
  

   centres 
  of 
  the 
  orbits, 
  multiply 
  F 
  x 
  by 
  the 
  cosine 
  of 
  the 
  angle 
  

   between 
  R 
  and 
  r. 
  The 
  cosine 
  of 
  this 
  angle 
  is 
  obtained 
  

   from 
  the 
  direct 
  product 
  of 
  R 
  and 
  r, 
  and 
  is 
  equal 
  to 
  

  

  -^—. 
  Hence 
  

   Rr 
  

  

  Fi=_ 
  Ki 
  (R 
  - 
  r)r 
  (14) 
  

  

  The 
  value 
  of 
  R.r 
  may 
  be 
  obtained 
  from 
  (7) 
  and 
  (9), 
  whence 
  

   H.Y 
  = 
  Ax 
  + 
  By 
  + 
  Cz 
  + 
  xD 
  eosu 
  + 
  zD 
  since 
  . 
  (15) 
  

  

  