﻿H. 
  A. 
  Newton 
  — 
  Capture 
  of 
  Comets 
  by 
  Planets. 
  191 
  

  

  ellipse. 
  The 
  ratio 
  of 
  the 
  axes 
  being 
  1 
  : 
  sin 
  0, 
  and 
  As 
  9 
  being 
  

   =mf, 
  the 
  area 
  of 
  the 
  ellipse 
  will 
  be 
  equal 
  to 
  

  

  s' 
  2 
  sin 
  

  

  : 
  ( 
  1 
  -( 
  cosfl 
  - 
  2 
  ^)> 
  

  

  18. 
  Maximum 
  action 
  of 
  the 
  planet. 
  — 
  For 
  two 
  particular 
  

   values 
  of 
  @ 
  the 
  isergonal 
  ellipses 
  become 
  points. 
  These 
  values 
  

   of 
  @ 
  result 
  if 
  the 
  maximum 
  effect 
  of 
  the 
  planet 
  in 
  increasing 
  

   and 
  in 
  decreasing 
  the 
  energy 
  of 
  the 
  comet 
  takes 
  place, 
  and 
  

   they 
  are 
  obtained 
  by 
  making 
  the 
  two 
  values 
  of 
  h 
  equal 
  to 
  each 
  

  

  . 
  As 
  . 
  

  

  other 
  in 
  (14), 
  that 
  is, 
  by 
  making 
  cos 
  6 
  — 
  =±1- 
  Since 
  at 
  

  

  the 
  same 
  time 
  h=2m@/s, 
  we 
  obtain 
  

  

  A 
  As 
  

  

  h 
  =. 
  j. 
  , 
  and 
  @ 
  = 
  -. 
  -x 
  . 
  . 
  (15) 
  

  

  cos 
  0=1= 
  l' 
  2m(cos0±l) 
  v 
  ; 
  

  

  Let 
  h' 
  and 
  h", 
  and 
  @! 
  and 
  @" 
  be 
  the 
  positive 
  and 
  negative 
  

   values 
  of 
  h 
  and 
  @ 
  in 
  (15) 
  and 
  we 
  may 
  construct 
  the 
  following 
  

   table 
  of 
  their 
  values. 
  As 
  in 
  Table 
  I 
  Jupiter 
  is 
  assumed 
  to 
  be 
  

   the 
  perturbing 
  planet. 
  

  

  Table 
  II. 
  

  

  0) 
  

  

  ft' 
  

  

  h" 
  

  

  @' 
  - 
  

  

  @" 
  

  

  0) 
  

  

  100° 
  

  

  h' 
  

  

  h" 
  

  

  @' 
  

  

  @" 
  

  

  0° 
  

  

  01443 
  

  

  a 
  

  

  3-14 
  

  

  , 
  — 
  a 
  

  

  •00426 
  - 
  

  

  00085 
  

  

  4-11 
  

  

  -0-83 
  

  

  10 
  

  

  01250 
  — 
  

  

  15174 
  

  

  3-04 
  

  

  -36 
  

  

  90 
  

  

  110 
  

  

  •00489 
  — 
  

  

  00072 
  

  

  5 
  

  

  12 
  

  

  -0 
  

  

  75 
  

  

  20 
  

  

  00927 
  - 
  

  

  03307 
  

  

  2-85 
  

  

  -10 
  

  

  15 
  

  

  120 
  

  

  •00598 
  - 
  

  

  00062 
  

  

  6 
  

  

  60 
  

  

  -0 
  

  

  68 
  

  

  30 
  

  

  00690 
  — 
  

  

  01290 
  

  

  2-69 
  

  

  - 
  5 
  

  

  03 
  

  

  130 
  

  

  00789 
  - 
  

  

  00055 
  

  

  9 
  

  

  09 
  

  

  -0 
  

  

  63 
  

  

  40 
  

  

  00544 
  — 
  

  

  00654 
  

  

  2-61 
  

  

  - 
  3 
  

  

  13 
  

  

  140 
  

  

  •01149 
  — 
  

  

  00050 
  

  

  13 
  

  

  71 
  

  

  -0 
  

  

  60 
  

  

  50 
  

  

  00457 
  |- 
  

  

  00387 
  

  

  2 
  61 
  

  

  — 
  2 
  

  

  21 
  

  

  150 
  

  

  •01934 
  - 
  

  

  00047 
  

  

  23 
  

  

  70 
  

  

  -0 
  

  

  57 
  

  

  60 
  

  

  0040? 
  - 
  

  

  00253 
  

  

  2-69 
  

  

  — 
  1 
  

  

  68 
  

  

  160 
  

  

  ■04192 
  - 
  

  

  00044 
  

  

  52 
  

  

  36 
  

  

  -0 
  

  

  55 
  

  

  70 
  

  

  00382 
  — 
  

  

  00179 
  

  

  2-86 
  

  

  - 
  1 
  

  

  34 
  

  

  170 
  

  

  •16336 
  — 
  

  

  00043 
  

  

  206-30 
  

  

  -0 
  

  

  54 
  

  

  80 
  

  

  00377 
  - 
  

  

  00134 
  

  

  3-14 
  

  

  - 
  1 
  

  

  11 
  

  

  180 
  

  

  a 
  — 
  

  

  00043 
  

  

  ex 
  

  

  -0-54 
  

  

  90 
  

  

  00390 
  ;— 
  

  

  00105 
  

  

  3-55 
  

  

  - 
  0-95 
  

  

  

  

  

  

  

  19. 
  Explanation 
  of 
  Table 
  II. 
  — 
  The 
  meaning 
  of 
  the 
  numbers 
  

   in 
  this 
  table 
  may 
  be 
  explained 
  by 
  an 
  example. 
  If 
  a 
  comet 
  

   moving 
  in 
  a 
  parabola 
  passes 
  near 
  to 
  Jupiter, 
  and 
  the 
  directions 
  

   of 
  the 
  two 
  original 
  motions 
  at 
  nearest 
  points 
  of 
  the 
  orbits 
  make 
  

   an 
  angle 
  of 
  10°, 
  then 
  the 
  greatest 
  action 
  of 
  Jupiter 
  (during 
  the 
  

   whole 
  period 
  of 
  transit) 
  in 
  diminishing 
  the 
  velocity 
  of 
  the 
  

   comet 
  in 
  its 
  orbit 
  about 
  the 
  sun 
  will 
  take 
  place 
  if 
  the 
  two 
  

   orbits 
  actually 
  intersect 
  {d—0), 
  and 
  if 
  the 
  comet 
  in 
  its 
  unper- 
  

   turbed 
  orbit 
  arrives 
  h'rst 
  at 
  the 
  point 
  of 
  intersection 
  at 
  the 
  

   instant 
  when 
  Jupiter 
  is 
  distant 
  therefrom 
  *01250 
  (the 
  earth's 
  

   mean 
  distance 
  from 
  the 
  sun 
  being 
  unity). 
  The 
  resulting 
  semi- 
  

   axis 
  major 
  of 
  the 
  comet's 
  orbit 
  about 
  the 
  sun 
  will 
  be 
  3'04. 
  

  

  On 
  the 
  other 
  hand, 
  the 
  greatest 
  effect 
  in 
  increasing 
  the 
  

   velocity 
  of 
  the 
  comet 
  will 
  take 
  place 
  when 
  the 
  two 
  orbits 
  

  

  