﻿H. 
  A. 
  Newton 
  — 
  Capture 
  of 
  Comets 
  by 
  Planets. 
  197 
  

  

  If 
  about 
  the 
  sun 
  as 
  a 
  center 
  a 
  sphere 
  S 
  be 
  described 
  with 
  

   an 
  arbitrary 
  radius 
  i\ 
  it 
  will 
  be 
  assumed 
  that 
  near 
  the 
  surface 
  

   of 
  6, 
  space 
  is 
  filled 
  equably 
  with 
  comets. 
  We 
  may 
  express 
  

   this 
  by 
  supposing 
  that 
  in 
  each 
  cubic 
  unit 
  of 
  space 
  near 
  0, 
  

   there 
  are 
  at 
  each 
  and 
  every 
  instant 
  n 
  comets. 
  As 
  the 
  orbits 
  

   are 
  all 
  assumed 
  to 
  be 
  parabolic, 
  the 
  n 
  comets 
  have 
  a 
  common 
  

   velocity 
  v. 
  

  

  Fig. 
  13; 
  w=120°. 
  

   Fig. 
  10; 
  w=50°. 
  Fig. 
  12; 
  «=60. 
  Fig. 
  11; 
  o=130. 
  

  

  H 
  — 
  .^ 
  

  

  

  

  

  

  

  

  

  

  

  

  

  

  \ 
  

  

  

  r 
  

  

  

  p 
  

  

  — 
  ^. 
  

  

  M 
  

  

  

  

  ~- 
  - 
  y~ 
  

  

  

  

  

  

  

  , 
  ..^p^ 
  

  

  

  ^1 
  

  

  Hi 
  

  

  1- 
  

  

  I 
  

  

  It 
  will 
  be 
  furthermore 
  assumed 
  that 
  the 
  directions 
  of 
  the 
  

   comets 
  in 
  each 
  cubic 
  unit 
  of 
  space 
  near 
  5 
  are 
  at 
  random, 
  that 
  

   is, 
  that 
  the 
  quits 
  and 
  goals 
  of 
  the 
  comet's 
  motions 
  relative 
  to 
  

   the 
  sun 
  are 
  distributed 
  equably 
  over 
  the 
  surface 
  of 
  the 
  celes- 
  

   tial 
  sphere. 
  

  

  27. 
  Number 
  of 
  comets 
  entering 
  %. 
  — 
  If 
  about 
  a 
  normal 
  to 
  S 
  

   as 
  an 
  axis 
  there 
  be 
  described 
  two 
  cones 
  cutting 
  the 
  celestial 
  

   sphere 
  in 
  two 
  small 
  circles 
  distant 
  from 
  the 
  point 
  where 
  the 
  

   normal 
  meets 
  the 
  celestial 
  sphere 
  ^r 
  and 
  -fy 
  + 
  d'fy, 
  then 
  of 
  the 
  n 
  

   comets 
  there 
  will 
  be 
  \n 
  sin 
  yjrdyjr 
  comets 
  whose 
  quits 
  are 
  

   between 
  the 
  two 
  circles. 
  Each 
  of 
  these 
  comets 
  will 
  move 
  per- 
  

   pendicularly 
  to 
  the 
  spherical 
  surface 
  5 
  with 
  the 
  velocity 
  

  

  