﻿184 
  H. 
  A. 
  Newton 
  — 
  Capture 
  of 
  Comets 
  oy 
  Planets. 
  

  

  2. 
  One 
  of 
  the 
  remarkable 
  distinctions 
  between 
  the 
  comets 
  

   of 
  long 
  (or 
  infinite) 
  periods, 
  and 
  those 
  of 
  short 
  periods, 
  is 
  that 
  

   the 
  orbits 
  of 
  the 
  latter 
  have 
  almost 
  without 
  exception 
  direct 
  

   motions 
  and 
  small 
  inclinations 
  to 
  the 
  plane 
  of 
  the 
  ecliptic, 
  

   while 
  the 
  orbits 
  of 
  the 
  former 
  have 
  all 
  possible 
  inclinations 
  

   between 
  0° 
  and 
  180°. 
  At 
  first 
  sight 
  this 
  seems 
  to 
  imply 
  that 
  

   the 
  two 
  groups 
  of 
  comets 
  are 
  radically 
  distinct 
  in 
  origin 
  or 
  

   nature 
  one 
  from 
  the 
  other. 
  The 
  most 
  natural 
  line 
  of 
  investi- 
  

   gation 
  therefore 
  is 
  the 
  effect 
  of 
  perturbations 
  in 
  bringing 
  or 
  

   uot 
  bringing 
  the 
  comets 
  to 
  move 
  with 
  the 
  planet 
  after 
  the 
  

   perturbation. 
  

  

  3. 
  The 
  algebraic 
  processes 
  by 
  which 
  was 
  obtained 
  the 
  form- 
  

   ula 
  for 
  the 
  change 
  of 
  energy 
  which 
  a 
  small 
  body 
  experiences 
  

   from 
  passing 
  near 
  a 
  planet 
  were 
  given 
  in 
  the 
  article 
  cited, 
  and 
  

   they 
  need 
  not 
  be 
  here 
  reproduced. 
  The 
  following 
  was 
  the 
  

   resulting 
  equation, 
  viz 
  : 
  

  

  kmfa~v, 
  cos 
  q> 
  sin 
  a 
  , 
  . 
  

  

  a 
  = 
  ^ 
  — 
  i 
  r 
  , 
  (i) 
  

  

  pv 
  Q 
  

  

  and 
  it 
  was 
  obtained 
  from 
  the 
  general 
  differential 
  equations 
  of 
  

   motion 
  by 
  making 
  assumptions 
  not 
  greatly 
  differing 
  from 
  those 
  

   used 
  in 
  obtaining 
  Laplace's 
  well 
  known 
  theorem, 
  that 
  a 
  sphere 
  

   of 
  suitable 
  magnitude 
  may 
  be 
  described 
  about 
  the 
  planet 
  as 
  a 
  

   center 
  and 
  that 
  for 
  a 
  tolerable 
  first 
  approximation 
  the 
  comet 
  

   may 
  be 
  regarded 
  as 
  moving 
  when 
  without 
  this 
  sphere 
  in 
  a 
  

   conic 
  section 
  of 
  which 
  the 
  sun 
  is 
  the 
  focus, 
  and 
  as 
  moving 
  

   when 
  within 
  the 
  sphere 
  in 
  a 
  conic 
  section 
  (an 
  hyperbola) 
  of 
  

   which 
  the 
  planet 
  is 
  the 
  focus. 
  In 
  other 
  words, 
  only 
  perturba- 
  

   tions 
  of 
  the 
  first 
  order 
  of 
  magnitude 
  are 
  taken 
  account 
  of. 
  A 
  

   comet 
  is 
  treated 
  throughout 
  this 
  paper 
  as 
  a 
  small 
  indivisible 
  

   body 
  whose 
  mass 
  may 
  be 
  neglected. 
  

  

  4. 
  Notation. 
  The 
  symbols 
  used 
  in 
  (1) 
  and 
  also 
  other 
  sym- 
  

   bols 
  which 
  I 
  shall 
  have 
  occasion 
  to 
  use 
  may 
  be 
  thus 
  defined. 
  

  

  Let 
  (ft 
  be 
  the 
  orbit 
  of 
  the 
  comet 
  about 
  the 
  sun 
  before 
  the 
  comet 
  

   comes 
  under 
  the 
  appreciable 
  action 
  of 
  the 
  planet 
  ; 
  

   (ft 
  the 
  orbit 
  of 
  the 
  comet 
  about 
  the 
  sun 
  after 
  perturbation 
  

   by 
  the 
  planet 
  ; 
  

  

  C 
  the 
  hyperbolic 
  orbit 
  of 
  the 
  comet 
  relative 
  to 
  Jupiter 
  when 
  

   near 
  the 
  planet 
  ; 
  

  

  J 
  the 
  elliptic 
  orbit 
  of 
  Jupiter 
  about 
  the 
  sun 
  ; 
  

  

  A 
  the 
  point 
  on 
  (ft 
  i 
  which 
  is 
  nearest 
  to 
  3 
  ; 
  

  

  E 
  the 
  point 
  on 
  jj 
  which 
  is 
  nearest 
  to 
  (ft/, 
  

  

  d 
  the 
  length 
  of 
  the 
  straight 
  line 
  EA 
  being 
  the 
  perpendicu- 
  

   lar 
  distance 
  between 
  the 
  orbits 
  at 
  their 
  nearest 
  ap- 
  

   proach 
  ; 
  

  

  gd 
  the 
  angle 
  between 
  the 
  tangent 
  of 
  (ft, 
  at 
  A 
  and 
  the 
  tangent 
  

   to 
  J 
  at 
  E 
  ; 
  

  

  