﻿H. 
  A. 
  Newton 
  — 
  Capture 
  of 
  Comets 
  by 
  Planets. 
  489 
  

  

  from 
  Jupiter's 
  quit, 
  while 
  only 
  38 
  of 
  them 
  have 
  quits 
  within 
  

   45° 
  of 
  Jupiter's 
  goal. 
  

  

  41. 
  Table 
  III 
  gives 
  the 
  distribution 
  of 
  the 
  comet 
  quits 
  rela- 
  

   tive 
  to 
  Jupiter's 
  quit. 
  It 
  may 
  also 
  be 
  used 
  to 
  determine 
  how 
  

   many 
  of 
  the 
  comets 
  whose 
  orbits 
  are 
  thus 
  changed 
  shall 
  have 
  

   an 
  inclination 
  to 
  the 
  plane 
  of 
  Jupiter's 
  orbit 
  less 
  than 
  a 
  given 
  

   angle. 
  

  

  Let 
  the 
  angle 
  be 
  30°. 
  Let 
  Q 
  be 
  Jupiter's 
  quit 
  on 
  the 
  celes- 
  

   tial 
  sphere, 
  Q' 
  the 
  comet's 
  quit 
  and 
  S 
  the 
  sun's 
  position 
  as 
  seen 
  

   from 
  Jupiter. 
  Then 
  in 
  the 
  triangle 
  QQ'S 
  put 
  a)" 
  for 
  QQ' 
  the 
  

   distance 
  of 
  the 
  quits. 
  The 
  side 
  QS 
  = 
  90°, 
  and 
  QSQ' 
  will 
  be 
  

   the 
  inclination 
  of 
  the 
  orbits. 
  Represent 
  this 
  angle 
  by 
  i 
  and 
  

   the 
  angle 
  Q'QS 
  by 
  q. 
  Then 
  sin 
  q 
  — 
  cot 
  to" 
  cot 
  i. 
  

  

  Let 
  two 
  small 
  circles 
  be 
  drawn 
  about 
  Q 
  at 
  distances 
  co" 
  and 
  

   o)"+clco 
  n 
  then 
  if 
  clco'' 
  be 
  made 
  15° 
  the 
  numbers 
  in 
  the 
  second 
  

   or 
  third 
  columns 
  of 
  table 
  III 
  indicate 
  how 
  many 
  quits 
  are 
  in 
  

   the 
  several 
  zones 
  of 
  15° 
  on 
  the 
  celestial 
  sphere. 
  These 
  may 
  

   be 
  distributed 
  at 
  smaller 
  intervals 
  than 
  15° 
  by 
  known 
  processes. 
  

   All 
  the 
  quits 
  that 
  lie 
  in 
  the 
  lime 
  between 
  two 
  semicircles 
  

   drawn 
  through 
  S 
  so 
  as 
  to 
  make 
  angles 
  of 
  30° 
  with 
  QS 
  will 
  evi- 
  

   dently 
  have 
  orbits 
  inclined 
  less 
  than 
  30° 
  to 
  Jupiter's 
  orbit. 
  

   From 
  <o" 
  = 
  to 
  to" 
  = 
  30° 
  all 
  the 
  quits 
  are 
  included 
  in 
  the 
  lune. 
  

   From 
  to" 
  ~ 
  80° 
  to 
  to" 
  = 
  90° 
  we 
  compute 
  yj 
  from 
  the 
  equation 
  

   sin 
  7] 
  = 
  cot 
  to" 
  cot 
  30° 
  ; 
  then 
  the 
  portion 
  of 
  the 
  quits 
  in 
  any 
  

   elemental 
  zone 
  that 
  fall 
  in 
  the 
  lune 
  is 
  to 
  the 
  whole 
  number 
  of 
  

   quits 
  in 
  that 
  elemental 
  zone 
  as 
  this 
  value 
  of 
  vj 
  is 
  to 
  90°. 
  These 
  

   may 
  be 
  summed 
  by 
  finite 
  summation, 
  and 
  the 
  result 
  is 
  that 
  

   among 
  the 
  839 
  comets 
  257 
  would 
  move 
  in 
  orbits 
  inclined 
  less 
  

   than 
  30° 
  to 
  the 
  orbit 
  of 
  Jupiter 
  

  

  42. 
  If 
  a 
  like 
  summation 
  be 
  made 
  for 
  the 
  equal 
  lune 
  that 
  

   contains 
  Jupiter's 
  goal 
  we 
  find 
  51 
  to 
  be 
  the 
  number 
  out 
  of 
  the 
  

   839 
  comets 
  which 
  move 
  in 
  orbits 
  inclined 
  more 
  than 
  150° 
  to 
  

   Jupiter's 
  orbit. 
  That 
  is, 
  somewhat 
  more 
  than 
  Jive 
  times 
  as 
  

   many 
  of 
  these 
  comets 
  move 
  in 
  direct 
  orbits 
  inclined 
  less 
  than 
  

   30° 
  to 
  Jupiter 
  '«§ 
  orbit 
  as 
  move 
  in 
  retrograde 
  orbits 
  inclined 
  less 
  

   than 
  30° 
  to 
  Jupitefs 
  orbit. 
  

  

  43. 
  The 
  comet 
  has 
  been 
  thus 
  far 
  considered 
  as 
  approaching 
  Ju- 
  

   piter 
  while 
  moving 
  in 
  a 
  parabolic 
  orbit 
  about 
  the 
  sun. 
  If 
  the 
  

   comet 
  however 
  is 
  moving 
  in 
  any 
  other 
  orbit, 
  and 
  it 
  passes 
  near 
  

   to 
  the 
  planet, 
  the 
  result 
  of 
  the 
  planet's 
  perturbing 
  action 
  will 
  

   in 
  general 
  be 
  quite 
  similar 
  to 
  the 
  result 
  when 
  the 
  orbit 
  is 
  para- 
  

   bolic, 
  the 
  other 
  circumstances 
  of 
  the 
  approach 
  being 
  ^assumed 
  

   to 
  be 
  alike 
  in 
  the 
  two 
  cases. 
  

  

  44. 
  These 
  are 
  perturbations 
  during 
  one 
  transit 
  past 
  the 
  planet. 
  

   But 
  the 
  comet, 
  unless 
  the 
  orbit 
  is 
  further 
  changed 
  by 
  another 
  

   planet, 
  must 
  return 
  at 
  each 
  revolution 
  to 
  the 
  place 
  where 
  it 
  

  

  