﻿190 
  H. 
  A. 
  Newton 
  — 
  Capture 
  of 
  Comets 
  oy 
  Planets. 
  

  

  and 
  (7) 
  are 
  used 
  in 
  making 
  the 
  table. 
  The 
  disturbing 
  planet 
  

   is 
  assumed 
  to 
  be 
  Jupiter, 
  so 
  that 
  m 
  was 
  taken 
  equal 
  to 
  1/1050 
  

   and 
  ^=5*2. 
  

  

  Table 
  I. 
  

  

  6) 
  

  

  d 
  

  

  5 
  

  

  A 
  

  

  0) 
  

  

  e 
  

  

  s 
  

  

  A 
  

  

  0° 
  

  

  0° 
  

  

  0' 
  

  

  0-414 
  

  

  •02886 
  

  

  100° 
  

  

  131° 
  

  

  48' 
  

  

  1-868 
  

  

  •00142 
  

  

  10 
  

  

  32 
  

  

  1 
  

  

  0-463 
  

  

  •02309 
  

  

  110 
  

  

  138 
  

  

  9 
  

  

  1-992 
  

  

  •00125 
  

  

  20 
  

  

  55 
  

  

  47 
  

  

  0-585 
  

  

  •01448 
  

  

  120 
  

  

  144 
  

  

  21 
  

  

  2-101 
  

  

  •00112 
  

  

  30 
  

  

  72 
  

  

  22 
  

  

  0-742 
  

  

  •00900 
  

  

  130 
  

  

  150 
  

  

  26 
  

  

  2-195 
  

  

  •00103 
  

  

  40 
  

  

  84 
  

  

  46 
  

  

  0-913 
  

  

  •00594 
  

  

  140 
  

  

  156 
  

  

  26 
  

  

  2-273 
  

  

  •00096 
  

  

  50 
  

  

  94 
  

  

  47 
  

  

  1-087 
  

  

  •00419 
  

  

  150 
  

  

  162 
  

  

  22 
  

  

  2-334 
  

  

  •00091 
  

  

  60 
  

  

  103 
  

  

  27 
  

  

  1-259 
  

  

  •00312 
  

  

  160 
  

  

  168 
  

  

  16 
  

  

  2379 
  

  

  •00088 
  

  

  70 
  

  

  111 
  

  

  14 
  

  

  1-426 
  

  

  •00244 
  

  

  170 
  

  

  174 
  

  

  8 
  

  

  2-405 
  

  

  •00086 
  

  

  80 
  

  

  118 
  

  

  27 
  

  

  1-584 
  

  

  •00197 
  

  

  180 
  

  

  180 
  

  

  

  

  2-414 
  

  

  •00085 
  

  

  90 
  

  

  125 
  

  

  16 
  

  

  1-732 
  

  

  •00165 
  

  

  

  

  

  

  

  15. 
  Using 
  these 
  values 
  of 
  6, 
  s 
  and 
  A 
  we 
  may 
  now 
  represent 
  

   graphically 
  the 
  dependence 
  of 
  @ 
  upon 
  the 
  other 
  two 
  variables 
  

   d 
  and 
  h 
  for 
  each 
  specified 
  value 
  of 
  co. 
  . 
  Let 
  d 
  and 
  h 
  be 
  Carte- 
  

   sian 
  coordinates, 
  then 
  for 
  each 
  point 
  of 
  the 
  coordinate 
  plane 
  

   there 
  is 
  a 
  value 
  of 
  @ 
  The 
  ambiguous 
  sign 
  will 
  be 
  fully 
  satis- 
  

   fied 
  by 
  giving 
  positive 
  and 
  negative 
  values 
  to 
  h. 
  For 
  an 
  

   assumed 
  value 
  of 
  @ 
  we 
  shall 
  have 
  a 
  curve 
  whose 
  equation 
  is 
  

   (13), 
  and 
  each 
  point 
  of 
  this 
  curve 
  represents 
  values 
  of 
  d 
  and 
  h 
  

   for 
  which 
  the 
  total 
  action 
  of 
  the 
  planet 
  upon 
  the 
  comet 
  will 
  

   be 
  to 
  reduce 
  the 
  energy 
  of 
  the 
  comet 
  a 
  constant 
  amount. 
  This 
  

   locus 
  will 
  be 
  called 
  an 
  isergonal 
  curve. 
  

  

  16. 
  Faisceau 
  of 
  isergonal 
  ellipses. 
  — 
  The 
  equation 
  (13) 
  of 
  the 
  

   isergonal 
  curve 
  may 
  be 
  written 
  

  

  4m@ 
  (A 
  cos 
  d 
  + 
  h 
  sin 
  2 
  6) 
  = 
  s(A 
  2 
  + 
  d' 
  2 
  + 
  h 
  2 
  sin 
  2 
  0), 
  

  

  and 
  this 
  is 
  the 
  equation 
  of 
  an 
  ellipse. 
  As 
  @ 
  changes 
  its 
  value 
  

   we 
  may 
  treat 
  it 
  as 
  a 
  parameter 
  and 
  we 
  have 
  a 
  faisceau 
  of 
  simi- 
  

   lar 
  isergonal 
  ellipses, 
  each 
  ellipse 
  symmetrical 
  with 
  the 
  axis 
  

   of 
  h. 
  The 
  radical 
  axis 
  of 
  the 
  faisceau 
  A 
  cos 
  0+ 
  h 
  sin 
  2 
  = 
  0, 
  

   and 
  the 
  imaginary 
  ellipse 
  A 
  3 
  +<$ 
  2 
  +A 
  2 
  sin 
  2 
  0=0, 
  are 
  theoretically 
  

   two 
  members 
  of 
  the 
  faisceau. 
  For 
  points 
  on 
  the 
  radical 
  axis 
  

   @— 
  a 
  and 
  therefore 
  for 
  this 
  locus 
  there 
  is 
  no 
  change 
  in 
  the 
  

   energy 
  of 
  the 
  comet. 
  

  

  17. 
  Center 
  and 
  area 
  of 
  the 
  isergonal 
  ellipse.— 
  -The 
  center 
  of 
  

   the 
  isergonal 
  ellipse 
  is 
  upon 
  the 
  axis 
  of 
  h 
  ; 
  making 
  d=0, 
  and 
  

   solving 
  for 
  h 
  we 
  have 
  

  

  h 
  = 
  

  

  2m@ 
  2m@ 
  , 
  

  

  ( 
  i 
  -H-sk)V 
  (]4) 
  

  

  The 
  first 
  term 
  of 
  the 
  second 
  member 
  of 
  (14) 
  is 
  the 
  ordinate 
  of 
  

   the 
  center, 
  and 
  the 
  second 
  term 
  is 
  the 
  semi-axis 
  major 
  of 
  the 
  

  

  