﻿H. 
  A. 
  Newton— 
  Capture 
  of 
  Comets 
  by 
  Planets. 
  185 
  

  

  Let 
  A 
  be 
  the 
  distance 
  which 
  the 
  planet 
  has 
  yet 
  to 
  pass 
  over 
  to 
  

   reach 
  E 
  when 
  the 
  comet 
  is 
  at 
  A 
  (h 
  may 
  be 
  negative) 
  ; 
  

  

  m 
  the 
  mass 
  of 
  the 
  planet, 
  sun's 
  mass=unity 
  ; 
  

  

  a 
  the 
  unit 
  of 
  distance, 
  in 
  general 
  the 
  mean 
  distance 
  of 
  the 
  

  

  earth 
  from 
  the 
  sun; 
  

   f 
  the 
  sun's 
  attractive 
  force 
  at 
  the 
  unit 
  of 
  distance 
  ; 
  

  

  v, 
  the 
  planet's 
  velocity 
  in 
  its 
  orbit 
  at 
  E 
  ; 
  

  

  v 
  o 
  the 
  comet's 
  velocity 
  in 
  its 
  orbit 
  C 
  when 
  the 
  comet 
  en- 
  

   ters 
  the 
  sphere 
  of 
  Jupiter's 
  perceptible 
  influence 
  ; 
  

  

  v 
  the 
  comet's 
  velocity 
  at 
  A 
  relative 
  to 
  the" 
  sun 
  ; 
  

  

  @ 
  t 
  the 
  semi-axis 
  major 
  of 
  (£", 
  (negative 
  if 
  QI 
  / 
  is 
  an 
  hyperbola) 
  

   @ 
  the 
  semi-axis 
  major 
  of 
  (£ 
  (negative 
  if 
  (£ 
  is 
  an 
  hyperbola) 
  

   p 
  the 
  perpendicular 
  from 
  the 
  planet 
  upon 
  asymptote 
  to 
  C 
  

   a 
  the 
  acute 
  angle 
  between 
  the 
  transverse 
  axis 
  of 
  C 
  and 
  

  

  the 
  asymptote 
  to 
  C. 
  

   cp 
  the 
  angle 
  between 
  the 
  tangent 
  to 
  jj 
  at 
  O 
  (drawn 
  in 
  the 
  

  

  direction 
  of 
  the 
  planet's 
  motion) 
  and 
  the 
  line 
  from 
  the 
  

  

  planet 
  to 
  the 
  vertices 
  and 
  center 
  of 
  C 
  ; 
  

   A 
  the 
  semi-transverse 
  axis 
  of 
  C 
  ; 
  

   B 
  the 
  semi-conjugate 
  axis 
  of 
  C 
  (hence 
  equal 
  to 
  jp) 
  ; 
  

   r 
  the 
  distance 
  of 
  the 
  planet 
  from 
  the 
  sun 
  ; 
  

   r 
  t 
  the 
  distance 
  of 
  the 
  comet 
  from 
  the 
  sun 
  ; 
  

   r 
  the 
  distance 
  of 
  the 
  comet 
  from 
  the 
  planet 
  ; 
  

   p 
  t 
  and 
  p 
  distances 
  of 
  the 
  comet 
  from 
  the 
  sun 
  at 
  selected 
  epochs 
  

  

  before 
  and 
  after 
  perturbation 
  ; 
  

  

  ^ 
  and 
  w 
  the 
  velocities 
  of 
  the 
  comet 
  at 
  the 
  selected 
  epochs 
  ; 
  

  

  , 
  . 
  ,. 
  , 
  2 
  2/a 
  2 
  2m/a 
  2 
  

  

  A 
  the 
  increase 
  to 
  which 
  v 
  — 
  - 
  — 
  — 
  — 
  - 
  — 
  receives 
  by 
  the 
  

  

  planet's 
  action 
  during 
  the 
  whole 
  period 
  in 
  which 
  the 
  

   comet 
  is 
  passing 
  near 
  to 
  Jupiter. 
  

  

  5. 
  If 
  we 
  assume 
  two 
  epochs, 
  one 
  before 
  and 
  one 
  after 
  the 
  

   perturbation, 
  at 
  which 
  the 
  comet 
  is 
  equally 
  distant 
  from 
  the 
  

   planet, 
  the 
  term 
  2mfa 
  2 
  /r 
  is 
  the 
  same 
  at 
  both 
  instants, 
  and 
  it 
  

   disappears 
  from 
  the 
  value 
  of 
  A. 
  Therefore 
  

  

  2/fc" 
  2 
  2/a* 
  

  

  A 
  z= 
  w 
  2 
  — 
  ^- 
  u 
  2 
  + 
  — 
  . 
  

  

  P 
  P, 
  

  

  But 
  by 
  the 
  well-known 
  formulas 
  from 
  the 
  law 
  of 
  gravitation, 
  

  

  < 
  = 
  ^ 
  - 
  w) 
  

  

  . 
  . 
  1 
  1 
  4m 
  cos 
  w 
  sin 
  a 
  

  

  that 
  is, 
  from 
  (1) 
  — 
  — 
  = 
  . 
  

  

  w 
  @ 
  @ 
  ps 
  

  

  