﻿188 
  II. 
  A. 
  Newton 
  — 
  Capture 
  of 
  Comets 
  by 
  Planets. 
  

  

  10. 
  To 
  find 
  a. 
  — 
  The 
  angle 
  a 
  is 
  the 
  acute 
  angle 
  between 
  the 
  

   asymptote 
  and 
  the 
  transverse 
  axis 
  of 
  the 
  hyperbola, 
  and 
  hence 
  

   from 
  the 
  nature 
  of 
  the 
  hyperbola 
  tan 
  a=B/A. 
  By 
  known 
  

   formulas 
  we 
  have, 
  if 
  the 
  planet 
  is 
  at 
  its 
  mean 
  distance 
  

  

  Therefore 
  -~ 
  = 
  — 
  , 
  or 
  A 
  = 
  — 
  r-. 
  

  

  v; 
  s 
  I 
  (V) 
  

  

  tt 
  B 
  p 
  s 
  2 
  (d 
  2 
  + 
  h 
  2 
  sin 
  2 
  6) 
  [ 
  K 
  } 
  

  

  Hence 
  from 
  (6) 
  tan 
  a 
  = 
  — 
  = 
  ^- 
  = 
  — 
  ± 
  -. 
  

  

  v 
  ' 
  A 
  A 
  mr 
  J 
  

  

  11. 
  To 
  find 
  <p. 
  — 
  The 
  orbit 
  of 
  the 
  comet 
  relative 
  to 
  Jupiter 
  

   lies 
  in 
  the 
  plane 
  YOB. 
  Let 
  i 
  be 
  the 
  inclination 
  of 
  the 
  plane 
  

   YOB 
  to 
  YOX, 
  measured 
  positive 
  from 
  x 
  positive 
  to 
  z 
  positive 
  ; 
  

   let 
  I 
  be 
  the 
  longitude 
  of 
  the 
  direction 
  YC, 
  measured 
  in 
  the 
  

   plane 
  YOX 
  from 
  OY, 
  that 
  is, 
  the 
  angle 
  made 
  by 
  YC 
  with 
  OY 
  

   produced 
  ; 
  let 
  \ 
  be 
  the 
  longitude 
  of 
  the 
  direction 
  YB 
  mea- 
  

   sured 
  in 
  the 
  plane 
  YOB 
  from 
  OY, 
  that 
  is, 
  the 
  angle 
  made 
  by 
  

   YB 
  with 
  OY 
  produced. 
  Imagine 
  now 
  a 
  sphere 
  described 
  

   about 
  Y 
  as 
  a 
  center 
  that 
  shall 
  cut 
  the 
  three 
  planes 
  XOY, 
  BOY 
  

   and 
  BCY 
  in 
  three 
  sides 
  of 
  a 
  right 
  angled 
  spherical 
  triangle. 
  

   The 
  hypotenuse 
  of 
  this 
  triangle 
  is 
  X, 
  the 
  base 
  I, 
  the 
  perpen- 
  

   dicular 
  \ir— 
  6, 
  and 
  the 
  angle 
  opposite 
  to 
  the 
  perpendicular 
  is 
  

   i 
  ; 
  hence 
  we 
  have 
  

  

  cos 
  X 
  = 
  cos 
  I 
  sin 
  0, 
  (8) 
  

  

  cos 
  = 
  sin 
  i 
  sin 
  A, 
  (9) 
  

  

  cot 
  i 
  = 
  sin 
  I 
  tan 
  6. 
  (10) 
  

   Also 
  from 
  the 
  triangles 
  OCY 
  and 
  BCY 
  

  

  tan 
  I 
  = 
  tan 
  OYC 
  = 
  - 
  ^ 
  = 
  - 
  -r—— 
  a 
  . 
  (11) 
  

  

  YC 
  h 
  tan 
  6 
  v 
  7 
  

  

  The 
  angle 
  <p 
  is 
  by 
  definition 
  the 
  angle 
  between 
  the 
  direction 
  

   OE, 
  and 
  a 
  line 
  in 
  the 
  plane 
  YOB 
  that 
  makes 
  with 
  YB 
  an 
  

   angle 
  a. 
  Hence 
  we 
  have 
  readily 
  

  

  cos 
  cp 
  = 
  sin 
  i 
  sin 
  (A 
  ± 
  oc). 
  (12) 
  

  

  These 
  equations 
  enable 
  us 
  to 
  compute 
  <p 
  in 
  terms 
  of 
  d, 
  h 
  and 
  

   co 
  ; 
  for 
  in 
  succession 
  may 
  be 
  computed 
  by 
  (3), 
  I 
  by 
  (11), 
  X 
  by 
  

   (8), 
  i 
  by 
  (10), 
  and 
  <p 
  by 
  (12). 
  

  

  12. 
  These 
  values 
  of 
  s, 
  p, 
  a 
  and 
  <p 
  give 
  by 
  equation 
  (2) 
  the 
  

   value 
  of 
  @. 
  The 
  suppositions 
  that 
  the 
  planet 
  is 
  at 
  its 
  mean 
  

   distance, 
  and 
  that 
  (JT 
  / 
  is 
  a 
  parabola, 
  are 
  involved 
  in 
  that 
  equa- 
  

   tion, 
  but 
  they 
  are 
  not 
  necessary 
  to 
  the 
  determination 
  of 
  @ 
  

   when 
  no 
  such 
  hypotheses 
  are 
  made, 
  and 
  changes 
  in 
  the 
  equation 
  

  

  