﻿H. 
  A. 
  Newton 
  — 
  Capture 
  of 
  Comets 
  by 
  Planets. 
  187 
  

  

  planet 
  to 
  move 
  from 
  O 
  towards 
  E 
  we 
  may 
  apply 
  an 
  equal, 
  

   opposite 
  motion 
  to 
  the 
  comet, 
  and 
  consider 
  the 
  planet 
  to 
  

   remain 
  at 
  rest 
  at 
  O. 
  Draw 
  AC 
  parallel 
  to 
  EO 
  and 
  make 
  AB 
  

   equal 
  to 
  the 
  distance 
  described 
  by 
  the 
  planet 
  during 
  the 
  time 
  

   that 
  the 
  comet 
  is 
  moving 
  from 
  Y 
  to 
  A. 
  Join 
  YB. 
  Then 
  

   since 
  YA 
  and 
  BA 
  represent 
  in 
  direction 
  and 
  magnitude 
  the 
  

   motions 
  of 
  the 
  two 
  bodies 
  in 
  a 
  given 
  interval, 
  the 
  third 
  side 
  

   YB 
  of 
  the 
  triangle 
  represents 
  in 
  magnitude 
  and 
  direction 
  the 
  

   motion 
  of 
  the 
  comet 
  relative 
  to 
  the 
  planet. 
  The 
  angle 
  YAB 
  

   is 
  the 
  angle 
  o>, 
  and 
  the 
  three 
  sides 
  of 
  the 
  triangle 
  YA, 
  YB 
  and 
  

   BA 
  are 
  proportional 
  to 
  v, 
  v 
  and 
  v 
  r 
  Let 
  the 
  angle 
  YBC 
  be 
  

   ; 
  then 
  from 
  the 
  triangle 
  YAB 
  we 
  have 
  

  

  v^z=.v 
  2 
  — 
  2v 
  t 
  v 
  COS 
  GQ 
  + 
  V% 
  

   and 
  . 
  v: 
  v 
  t 
  \ 
  v 
  o 
  :: 
  sin 
  ; 
  sin 
  (6— 
  go) 
  : 
  sin 
  go. 
  (3) 
  

  

  Since 
  v 
  and 
  v 
  / 
  can 
  be 
  computed 
  from 
  the 
  given 
  elements 
  of 
  

   the 
  orbits 
  of 
  the 
  planet 
  and 
  comet, 
  we 
  may 
  readily 
  compute 
  

   from 
  co 
  the 
  value 
  of 
  s, 
  or 
  v 
  Q 
  /v 
  r 
  But 
  if 
  the 
  planet 
  is 
  at 
  its 
  mean 
  

   distance 
  from 
  the 
  sun, 
  and 
  the 
  comet's 
  orbit 
  is 
  parabolic, 
  

   v' 
  2 
  = 
  2v/, 
  and 
  we 
  have 
  

  

  s 
  2 
  = 
  3 
  — 
  2^2 
  cos 
  go. 
  (4) 
  

  

  Also 
  from 
  the 
  triangle 
  

  

  2v* 
  =,vr 
  Q 
  * 
  -t- 
  2v 
  / 
  y 
  / 
  cos 
  G 
  + 
  v*, 
  

   or 
  2s 
  cos 
  6 
  = 
  1 
  — 
  s\ 
  (5) 
  

  

  9. 
  To 
  find 
  p. 
  — 
  The 
  planet 
  being 
  regarded 
  at 
  rest 
  at 
  O 
  and 
  

   the 
  relative 
  unperturbed 
  motion 
  of 
  the 
  comet 
  being 
  along 
  YB, 
  

   this 
  line 
  may 
  within 
  admissible 
  limits 
  of 
  error 
  be 
  treated 
  as 
  

   one 
  asymptote 
  of 
  the 
  relative 
  orbit 
  C. 
  The 
  perpendicular 
  

   from 
  O 
  upon 
  YB 
  will 
  then 
  be 
  by 
  definition 
  (Art. 
  4) 
  the 
  line 
  

   p. 
  Draw 
  OX 
  from 
  O 
  perpendicular 
  to 
  OY 
  and 
  OE, 
  and 
  let 
  

   these 
  three 
  lines 
  be 
  coordinate 
  axes. 
  Let 
  the 
  line 
  AB 
  meet 
  

   the 
  plane 
  XOY 
  in 
  C. 
  Join 
  OC, 
  let 
  fall 
  OD 
  perpendicular 
  to 
  

   YB, 
  and 
  join 
  CD. 
  Since 
  EA 
  is 
  perpendicular 
  to 
  AY 
  and 
  also 
  

   to 
  EO, 
  and 
  so 
  to 
  its 
  parallel 
  line 
  AC, 
  therefore 
  it 
  is 
  perpen- 
  

   dicular 
  to 
  the 
  plane 
  YAC. 
  Hence 
  OC, 
  parallel 
  to 
  EA 
  is 
  per- 
  

   pendicular 
  to 
  the 
  plane, 
  and 
  so 
  perpendicular 
  to 
  CD. 
  Again 
  

   CDY 
  is 
  a 
  right 
  angle; 
  for 
  OD 
  2 
  + 
  DY 
  2 
  =OY 
  2 
  = 
  OC 
  2 
  + 
  CY 
  2 
  , 
  and 
  

   OD 
  2 
  = 
  OC 
  2 
  -hDC 
  2 
  . 
  Hence 
  DC 
  2 
  +DY 
  2 
  =CY 
  a 
  , 
  and 
  consequently 
  

   CDY 
  is 
  a 
  right 
  angle. 
  

  

  The 
  quantity 
  h 
  is 
  the 
  line 
  BC 
  ; 
  for 
  h 
  is 
  the 
  distance 
  which 
  

   the 
  planet, 
  when 
  the 
  comet 
  is 
  at 
  A, 
  has 
  yet 
  to 
  pass 
  over 
  before 
  

   reaching 
  E. 
  But 
  the 
  comet 
  was 
  at 
  Y 
  when 
  the 
  planet 
  was 
  at 
  

   O, 
  and 
  the 
  planet 
  describes 
  B 
  A, 
  while 
  the 
  comet 
  describes 
  YA, 
  

   leaving 
  BC 
  as 
  the 
  distance 
  yet 
  to 
  be 
  described 
  or 
  h. 
  But 
  the 
  

   angle 
  CBD 
  is 
  6, 
  so 
  that 
  we 
  have 
  

  

  p* 
  = 
  OD 
  2 
  = 
  OC 
  2 
  + 
  CD 
  2 
  = 
  d 
  2 
  + 
  A 
  2 
  sin 
  2 
  d. 
  (6) 
  

  

  