﻿H. 
  A. 
  Newton 
  — 
  Capture 
  of 
  Comets 
  by 
  Planets. 
  485 
  

  

  35. 
  If 
  Jupiter 
  is 
  the 
  planet, 
  m 
  = 
  1 
  J$ 
  , 
  and 
  we 
  may 
  express 
  

   these 
  ratios 
  as 
  

  

  1 
  000 
  000 
  000 
  : 
  126 
  : 
  839 
  : 
  1701 
  : 
  2670. 
  

  

  That 
  is, 
  assuming 
  the 
  hypotheses 
  of 
  Art. 
  26, 
  and 
  regarding 
  the 
  

   planet 
  as 
  without 
  dimension 
  so 
  as 
  to 
  intercept 
  any 
  comets, 
  if 
  

   in 
  a 
  given 
  period 
  of 
  time 
  a 
  thousand 
  million 
  comets 
  come 
  in 
  

   parabolic 
  orbits 
  nearer 
  to 
  the 
  sun 
  than 
  Jupiter 
  ; 
  126 
  of 
  them 
  

   will 
  have 
  their 
  orbits 
  changed 
  into 
  ellipses 
  with 
  periodic 
  times 
  

   less 
  than 
  one-half 
  that 
  of 
  Jitpiter 
  ; 
  889 
  of 
  them 
  will 
  have 
  their 
  

   orbits 
  changed 
  into 
  ellipses 
  with 
  periodic 
  times 
  less 
  than 
  that 
  

   of 
  Jupiter 
  ; 
  1701 
  of 
  them 
  will 
  have 
  their 
  orbits 
  changed 
  into 
  

   ellipses 
  with 
  periodic 
  times 
  less 
  than 
  once 
  and 
  a 
  half 
  times 
  

   that 
  of 
  Jupiter 
  ; 
  and 
  2670 
  of 
  them 
  will 
  have 
  their 
  orbits 
  

   changed 
  into 
  ellipses 
  with 
  periodic 
  times 
  less 
  than 
  twice 
  that 
  of 
  

   Jupiter. 
  

  

  36. 
  Another 
  and 
  perhaps 
  a 
  more 
  important 
  inquiry 
  is 
  this, 
  

   what 
  effect 
  have 
  the 
  perturbations 
  of 
  the 
  planet 
  in 
  bringing 
  or 
  

   not 
  bringing 
  the 
  comets 
  to 
  move 
  in 
  the 
  same 
  direction 
  that 
  

   the 
  planet 
  is 
  moving 
  after 
  the 
  comets 
  have 
  by 
  perturbation 
  had 
  

   their 
  periodic 
  times 
  largely 
  reduced. 
  For 
  simplicity 
  and 
  as 
  a 
  

   special 
  example 
  I 
  shall 
  consider 
  the 
  action 
  of 
  Jupiter 
  only, 
  

   and 
  also 
  only 
  his 
  action 
  upon 
  those 
  comets 
  whose 
  periodic 
  

   times 
  are 
  reduced 
  to 
  be 
  less 
  than 
  Jupiter's 
  period, 
  the 
  original 
  

   orbits 
  of 
  the 
  comets 
  being 
  parabolic. 
  In 
  other 
  words, 
  how 
  

   many 
  of 
  the 
  839 
  comets 
  which 
  are 
  reduced 
  (Art. 
  35,) 
  to 
  have 
  

   periodic 
  times 
  less 
  than 
  Jupiter's 
  period 
  will 
  after 
  perturbation 
  

   have 
  goals 
  distant 
  less 
  than 
  15°, 
  30°, 
  45°. 
  etc., 
  severally 
  from 
  

   Jupiter's 
  goal? 
  

  

  37. 
  Let 
  BA, 
  Fig. 
  19, 
  be 
  drawn 
  to 
  represent 
  v 
  x 
  and 
  CA 
  to 
  

   represent 
  v 
  iy 
  /2. 
  With 
  A 
  as 
  a 
  center 
  and 
  AB 
  and 
  AC 
  as 
  

   radii 
  describe 
  the 
  semicircumferences 
  BLO 
  and 
  CHGr. 
  Let 
  

   the 
  angle 
  BAH 
  be 
  made 
  equal 
  to 
  co 
  and 
  BH 
  be 
  drawn 
  ; 
  then 
  

   HA 
  will 
  represent 
  the 
  comet's 
  velocity 
  about 
  the 
  sun, 
  BA 
  the 
  

   planets 
  velocity 
  about 
  the 
  sun, 
  and 
  therefore 
  HB 
  the 
  comet's 
  

   velocity 
  v 
  in 
  its 
  orbit 
  about 
  the 
  planet 
  before 
  perturbation. 
  

   About 
  B 
  as 
  center 
  describe 
  the 
  semicircumference 
  KHT. 
  

   Since 
  the 
  relative 
  velocity 
  after 
  as 
  well 
  as 
  before 
  perturbation 
  

   is 
  equal 
  to 
  HB, 
  therefore 
  the 
  velocity 
  of 
  the 
  comet 
  about 
  the 
  

   sun 
  after 
  perturbation 
  will 
  evidently 
  be 
  represented 
  by 
  a 
  line 
  

   drawn 
  from 
  some 
  point 
  in 
  the 
  semicircumference 
  KHT 
  to 
  A. 
  

   If 
  the 
  velocity 
  is 
  increased 
  the 
  new 
  velocity 
  will 
  be 
  represented 
  

   by 
  a 
  line 
  to 
  A 
  from 
  some 
  point 
  in 
  the 
  arc 
  KH, 
  if 
  diminished 
  

   by 
  a 
  line 
  to 
  A 
  from 
  some 
  point 
  in 
  the 
  arc 
  HT. 
  If 
  the 
  new 
  

   velocity 
  is 
  less 
  than 
  the 
  planet's 
  velocity, 
  and 
  so 
  the 
  new 
  com- 
  

   etic 
  period 
  less 
  than 
  the 
  planet's 
  period, 
  the 
  new 
  velocity 
  will 
  

   be 
  represented 
  by 
  a 
  line 
  to 
  A 
  from 
  some 
  point 
  in 
  the 
  arc 
  ET. 
  

  

  