﻿186 
  H. 
  A. 
  Newton 
  — 
  Capture 
  of 
  Comets 
  by 
  Planets. 
  

  

  This 
  equation 
  is 
  valid 
  whatever 
  be 
  @, 
  the 
  major 
  axis 
  of 
  the 
  

   orbit 
  (E,, 
  and 
  may 
  be 
  used 
  to 
  determine 
  the 
  major 
  axis 
  of 
  

   either 
  orbit 
  from 
  the 
  elements 
  of 
  the 
  other. 
  My 
  present 
  pur- 
  

   pose 
  is, 
  however, 
  to 
  study 
  the 
  action 
  of 
  Jupiter 
  in 
  changing 
  

   orbits 
  that 
  are 
  originally 
  parabolas, 
  and 
  hence 
  in 
  general 
  @ 
  / 
  

   will 
  be 
  taken 
  infinite, 
  fn 
  that 
  case 
  

  

  PS 
  (2) 
  

  

  4w 
  cos 
  cp 
  sin 
  a 
  

  

  It 
  will 
  be 
  found 
  that 
  the 
  second 
  number 
  of 
  (2) 
  depends 
  on 
  

   a), 
  d 
  and 
  A, 
  and 
  these 
  are 
  known 
  quantities 
  when 
  the 
  elements 
  

   of 
  (£/ 
  and 
  3 
  are 
  given. 
  The 
  use 
  of 
  the 
  equation 
  is 
  moreover 
  

   greatly 
  simplified 
  and 
  enhanced 
  by 
  the 
  fact 
  that 
  the 
  plane 
  of 
  

   the 
  planet's 
  orbit 
  is 
  involved 
  only 
  in 
  so 
  far 
  as 
  that 
  it 
  must 
  

   contain 
  the 
  tangent 
  to 
  M 
  at 
  E. 
  

  

  6. 
  In 
  the 
  second 
  member 
  of 
  (2) 
  all 
  the 
  factors 
  are 
  positive 
  

   except 
  cos 
  <p, 
  hence, 
  if 
  ^<-§7r, 
  @ 
  is 
  positive 
  and 
  the 
  orbit 
  (£ 
  is 
  

   an 
  ellipse 
  ; 
  but, 
  if 
  ^C>§7r, 
  @ 
  is 
  negative 
  and 
  (£ 
  is 
  an 
  hyperbola. 
  

   This 
  result 
  may 
  be 
  thus 
  expressed 
  ; 
  if 
  the 
  comet 
  passes 
  in 
  

   front 
  of 
  Jupiter 
  the 
  kinetic 
  energy 
  of 
  the 
  comet 
  is 
  diminished, 
  

   if 
  it 
  passes 
  behind 
  the 
  planet 
  the 
  kinetic 
  energy 
  of 
  the 
  comet 
  is 
  

   increased. 
  The 
  reason 
  for 
  this 
  may 
  also 
  be 
  given 
  in 
  general 
  

   language. 
  If 
  the 
  comet 
  passes 
  in 
  front 
  of 
  the 
  planet 
  the 
  

   comet's 
  attraction 
  increases 
  the 
  velocity, 
  and 
  hence 
  increases 
  

   the 
  kinetic 
  energy 
  of 
  the 
  planet 
  and 
  vice 
  versa. 
  But 
  the 
  total 
  

   energy 
  of 
  the 
  two 
  bodies 
  is 
  constant 
  ; 
  so 
  that 
  when 
  that 
  of 
  the 
  

   planet 
  is 
  increased, 
  that 
  of 
  the 
  comet 
  is 
  diminished 
  and 
  vice 
  

   versa. 
  

  

  Y. 
  It 
  is 
  desirable 
  now 
  to 
  transform 
  the 
  value 
  of 
  @ 
  given 
  in 
  

   equation 
  (2) 
  so 
  as 
  to 
  be 
  able 
  to 
  determine 
  the 
  major 
  axis 
  of 
  the 
  

   new 
  orbit 
  of 
  the 
  comet 
  directly 
  from 
  the 
  circumstances 
  of 
  its 
  

   initial 
  approach 
  to 
  the 
  planet 
  before 
  perturbation 
  ; 
  in 
  other 
  

   words, 
  to 
  find 
  @ 
  in 
  terms 
  of 
  &>, 
  d 
  and 
  h. 
  For 
  this 
  we 
  must 
  

   find 
  in 
  terms 
  of 
  co, 
  d 
  and 
  A, 
  values 
  for 
  s, 
  p, 
  a 
  and 
  <p. 
  

  

  8. 
  To 
  find 
  s. 
  — 
  In 
  fig. 
  1 
  let 
  A 
  and 
  E 
  repre- 
  

   sent 
  the 
  two 
  points 
  A 
  and 
  E 
  as 
  defined 
  above 
  

   (Art. 
  4), 
  and 
  the 
  line 
  AE 
  represent 
  d. 
  Let 
  

   AY 
  be 
  the 
  tangent 
  to 
  (H 
  / 
  at 
  A, 
  and 
  EO 
  the 
  

   tangent 
  to 
  J 
  at 
  E. 
  It 
  is 
  an 
  admissible 
  sup- 
  

   position 
  that 
  the 
  planet 
  is 
  describing 
  the 
  

   straight 
  line 
  OE, 
  and 
  that 
  the 
  comet 
  in 
  its 
  

   unperturbed 
  orbit 
  is 
  describing 
  the 
  straight 
  

   line 
  YA. 
  At 
  some 
  certain 
  moment 
  the 
  line 
  

   joining 
  the 
  planet 
  and 
  the 
  unperturbed 
  comet 
  

   must 
  evidently 
  be 
  perpendicular 
  to 
  OE. 
  Let 
  

   OY 
  be 
  the 
  line 
  joining 
  the 
  bodies 
  at 
  that 
  

   moment, 
  so 
  that 
  the 
  planet 
  is 
  at 
  O 
  when 
  the 
  comet 
  is 
  at 
  Y, 
  

   and 
  EOY 
  is 
  a 
  right 
  angle. 
  Instead, 
  however, 
  of 
  supposing 
  the 
  

  

  