﻿G6 
  Dr. 
  A. 
  C. 
  Oreiiore 
  on 
  the 
  Formation 
  of 
  the 
  

  

  Average 
  Values, 
  

  

  The 
  average 
  values 
  of 
  the 
  forces 
  along 
  and 
  perpendicular 
  

   to 
  r 
  are 
  obtained 
  from 
  (28) 
  and 
  (3:5) 
  respectively 
  by 
  

   integration, 
  and 
  we 
  find, 
  putting 
  r 
  = 
  av; 
  a=ma 
  1t 
  ; 
  a' 
  = 
  na# 
  ; 
  

   and 
  A 
  = 
  (i' 
  2 
  -f 
  ??i 
  2 
  + 
  n" 
  2 
  ) 
  - 
  * 
  and 
  including 
  all 
  terms 
  in 
  v 
  -6 
  , 
  

  

  F 
  1= 
  - 
  ^=2 
  ] 
  l-t-3mn(cos 
  7 
  W 
  + 
  7-5™ 
  2 
  n 
  2 
  (cos 
  2 
  7 
  )A 
  4 
  + 
  3-75(m 
  2 
  +n 
  2 
  

  

  along 
  lvci 
  # 
  C. 
  

  

  — 
  2mn 
  cos 
  y)(co± 
  2 
  \)u 
  2 
  A 
  4 
  -\- 
  2 
  Cv 
  25 
  rnn 
  cos 
  y(in 
  2 
  -\-n 
  2 
  

  

  — 
  2mn 
  cos 
  7) 
  (cos 
  2 
  A> 
  2 
  A 
  6 
  + 
  14' 
  7656(??i 
  2 
  + 
  ?i 
  2 
  

  

  — 
  2m/i 
  cos 
  7) 
  2 
  (cos 
  4 
  A 
  ) 
  v 
  4 
  A 
  8 
  — 
  1 
  • 
  5(m 
  2 
  4- 
  n 
  2 
  — 
  2 
  ra?i 
  cos 
  7) 
  

   (cos 
  2 
  X) 
  A 
  2 
  — 
  5m>i 
  cos 
  <y(in* 
  + 
  n 
  2 
  — 
  2mn 
  cos 
  7) 
  (cos 
  2 
  X) 
  A 
  4 
  

   -•875(m 
  2 
  + 
  n 
  2 
  -2mn 
  cos 
  7) 
  2 
  (cos 
  4 
  X)tM 
  6 
  + 
  }• 
  (34) 
  

  

  F 
  2 
  =-/3/3'(cos 
  7 
  )F 
  1 
  (35) 
  

  

  along 
  along 
  

  

  Fj 
  = 
  rtan 
  X) 
  tt 
  — 
  21 
  l*5(m 
  2 
  + 
  w 
  2 
  — 
  2wmcos7)(cos 
  2 
  X) 
  A 
  2 
  -j- 
  5mn 
  cos 
  y(m?-\ 
  n 
  2 
  

   perp. 
  ivtl^ 
  L 
  

  

  — 
  2mn 
  cos 
  7 
  )(cos 
  2 
  X)A 
  4 
  + 
  -875(wi 
  2 
  + 
  n 
  2 
  

   _2mrccos7) 
  2 
  (cos 
  4 
  X> 
  2 
  A 
  6 
  + 
  1. 
  . 
  . 
  . 
  (3G) 
  

  

  F 
  2 
  =-^'(cos7)Fi. 
  • 
  (37) 
  

  

  perp. 
  P 
  e 
  'T- 
  

  

  Equations 
  (34) 
  to 
  (37) 
  give 
  the 
  average 
  values 
  of 
  the 
  

   first 
  and 
  second 
  component 
  forces 
  resolved 
  along 
  and 
  per- 
  

   pendicular 
  to 
  r 
  in 
  the 
  meridian 
  plane, 
  but 
  it 
  remains 
  to 
  be 
  

   shown 
  that 
  the 
  force 
  perpendicular 
  to 
  both 
  of 
  these 
  directions, 
  

   namely, 
  along 
  a 
  tangent 
  line 
  to 
  the 
  circle 
  of 
  latitude 
  at 
  the 
  

   point 
  where 
  the 
  second 
  orbit 
  is 
  situated, 
  is 
  zero, 
  and 
  con- 
  

   sequently 
  that 
  the 
  forces 
  above 
  given 
  are 
  sufficient 
  to 
  

   determine 
  equilibrium 
  conditions. 
  The 
  direction 
  along 
  which 
  

   we 
  are 
  to 
  resolve 
  the 
  forces 
  is 
  parallel 
  to 
  the 
  j-axis, 
  and 
  the 
  

   cosine 
  of 
  the 
  angle 
  between 
  R, 
  along 
  which 
  the 
  force 
  nets, 
  

  

  and 
  the 
  j-ax's, 
  along 
  which 
  we 
  are 
  to 
  resolve 
  it, 
  is 
  -^-. 
  

   By 
  (9) 
  when 
  y 
  = 
  0, 
  we 
  find 
  that 
  

  

  R_.,7 
  _ 
  B 
  _ 
  -aC 
  + 
  a'C 
  

  

  