﻿H. 
  A. 
  Newton 
  — 
  Capture 
  of 
  Comets 
  by 
  Planets. 
  483 
  

  

  comets 
  if 
  unperturbed. 
  Evidently 
  an 
  equal 
  number 
  cross 
  the 
  

   surface 
  S' 
  entering 
  the 
  sphere 
  in 
  each 
  unit 
  of 
  time. 
  

  

  If 
  now 
  o) 
  be 
  the 
  angle 
  which 
  the 
  comet's 
  unperturbed 
  

   motion 
  is 
  making 
  with 
  the 
  planet's 
  motion, 
  and 
  if 
  v 
  1 
  or 
  its 
  

   equal 
  v/ 
  y/%, 
  be 
  the 
  planet's 
  velocity 
  in 
  its 
  orbit 
  about 
  the 
  sun, 
  

   then 
  v* 
  = 
  £#[3 
  — 
  2 
  4/ 
  2 
  cos 
  to]. 
  The 
  element 
  dS 
  may 
  be 
  taken 
  

   to 
  be 
  the 
  elemental 
  zone 
  between 
  the 
  two 
  small 
  circles 
  whose 
  

   common 
  pole 
  is 
  the 
  planet's 
  quit, 
  and 
  whose 
  distances 
  . 
  from 
  

   the 
  planet's 
  quit 
  are 
  co 
  and 
  co 
  + 
  dco. 
  Then 
  d$ 
  = 
  27rsin<:i> 
  dco. 
  

   The 
  number 
  of 
  comets 
  entering 
  S' 
  in 
  a 
  unit 
  of 
  time 
  with 
  quits 
  

   within 
  that 
  elemental 
  zone 
  will 
  be 
  

  

  invjr'* 
  X 
  2 
  n 
  sin 
  go 
  doo 
  = 
  ^ 
  (3 
  — 
  2 
  a/2 
  cos 
  go) 
  sin 
  go 
  dGO. 
  

  

  2^/2 
  

  

  The 
  integral 
  of 
  this, 
  

  

  Tinvr 
  

  

  — 
  / 
  (3 
  — 
  2 
  ,y/2 
  cos 
  go) 
  " 
  sin 
  go 
  dco 
  = 
  ^nnvr' 
  9 
  , 
  

  

  expresses 
  the 
  total 
  number 
  of 
  comets 
  that, 
  under 
  the 
  hypothe- 
  

   ses 
  that 
  have 
  been 
  made, 
  would 
  in 
  a 
  unit 
  of 
  time 
  enter 
  the 
  

   sphere 
  S 
  r 
  . 
  

  

  31. 
  If 
  we 
  compare 
  the 
  two 
  expressions 
  obtained 
  in 
  Arts. 
  27 
  

   and 
  30 
  we 
  find 
  that 
  the 
  number 
  of 
  comets 
  which, 
  in 
  a 
  given 
  

   period 
  of 
  time 
  come 
  nearer 
  to 
  the 
  sun 
  than 
  r 
  is 
  to 
  the 
  number 
  

   that 
  (unperturbed) 
  come 
  nearer 
  to 
  the 
  planet 
  than 
  r' 
  as 
  6r 
  2 
  is 
  

   to 
  7r 
  /2 
  . 
  The 
  factor 
  -J 
  expresses 
  the 
  increase 
  of 
  numbers 
  caused 
  

   by 
  the 
  planet's 
  motion 
  in 
  its 
  circular 
  orbit. 
  The 
  value 
  of 
  r\ 
  

   as 
  has 
  been 
  said, 
  must 
  not 
  be 
  too 
  small, 
  nor 
  yet 
  must 
  it 
  be 
  very 
  

   large. 
  

  

  82. 
  In 
  order 
  to 
  determine 
  the 
  number 
  N 
  of 
  comets 
  which 
  in 
  

   a 
  unit 
  of 
  time 
  will 
  have 
  their 
  periodic 
  times 
  reduced 
  below 
  a 
  

   given 
  period 
  we 
  may 
  make 
  use 
  of 
  the 
  isergonal 
  curves 
  repre- 
  

   sented 
  in 
  Figs. 
  2-18. 
  Although 
  the 
  diagrams 
  were 
  not 
  con- 
  

   structed 
  to 
  exhibit 
  the 
  motions 
  of 
  the 
  bodies, 
  yet 
  they 
  may 
  be 
  

   utilized 
  for 
  that 
  purpose. 
  Let 
  OH 
  be 
  the 
  tangent 
  to 
  the 
  

   planet's 
  orbit, 
  O 
  the 
  place 
  of 
  the 
  planet 
  considered 
  at 
  rest, 
  and 
  

   let 
  the 
  plane 
  HOE 
  contain 
  the 
  shortest 
  line 
  d 
  between 
  the 
  

   two 
  orbits. 
  This 
  d 
  will 
  be 
  the 
  abscissa 
  of 
  the 
  point 
  at 
  which 
  

   the 
  comet's 
  unperturbed 
  orbit 
  will 
  cut 
  the 
  plane. 
  The 
  ordi- 
  

   nate 
  of 
  the 
  same 
  point, 
  produced 
  if 
  necessary, 
  will 
  be 
  the 
  pro- 
  

   jection 
  of 
  the 
  comet's 
  path 
  upon 
  the 
  plane 
  HOE, 
  and 
  the 
  

   comet's 
  path 
  makes 
  with 
  the 
  plane 
  the 
  angle 
  6. 
  The 
  velocity 
  

   of 
  the 
  comet 
  perpendicular 
  to 
  the 
  plane 
  will 
  be 
  v 
  sin 
  6. 
  By 
  

   reason 
  of 
  the 
  hypothesis 
  that 
  the 
  comets 
  are 
  equably 
  distribu- 
  

   ted, 
  the 
  points 
  of 
  intersection 
  with 
  the 
  plane 
  HOE 
  will 
  be 
  

   equably 
  distributed 
  over 
  the 
  plane. 
  Hence 
  the 
  number 
  of 
  

  

  