﻿H. 
  A. 
  Newton 
  — 
  Capture 
  of 
  Comets 
  by 
  Planets. 
  189 
  

  

  that 
  are 
  not 
  serious 
  would 
  make 
  it 
  applicable 
  without 
  these 
  

   limitations. 
  The 
  quantities 
  in 
  the 
  several 
  equations 
  may 
  be 
  

   regarded 
  as 
  having 
  values 
  : 
  — 
  

  

  cl 
  positive, 
  

  

  h 
  positive 
  or 
  negative, 
  

   a 
  positive 
  and 
  less 
  than 
  \n, 
  

   od, 
  6, 
  q) 
  and 
  i 
  positive 
  and 
  less 
  than 
  n, 
  

   I 
  and 
  A 
  positive 
  and 
  less 
  than 
  In. 
  

  

  13. 
  We 
  may, 
  however, 
  also 
  find 
  directly 
  the 
  value 
  of 
  @ 
  in 
  

   terms 
  of 
  d, 
  h, 
  and 
  the 
  known 
  functions 
  of 
  ft). 
  

   From 
  (12) 
  

  

  cos 
  cp 
  sin 
  a 
  = 
  sin 
  i 
  sin 
  X 
  cos 
  a 
  sin 
  a^z 
  sin 
  i 
  cos 
  X 
  sin 
  2 
  a. 
  

  

  From 
  (7) 
  

  

  AB 
  ' 
  . 
  B 
  9 
  

  

  cos 
  a 
  sin 
  a 
  = 
  -r~r 
  — 
  _„, 
  and 
  sin 
  a 
  = 
  -— 
  - 
  — 
  — 
  . 
  

   A+B 
  J 
  ' 
  A- 
  + 
  B" 
  

  

  From 
  (10) 
  and 
  (8) 
  

  

  sin 
  i 
  cos 
  X 
  

  

  = 
  ± 
  

  

  cos 
  I 
  sin 
  

  

  e 
  

  

  cot 
  £ 
  sin 
  

  

  =fc 
  

  

  

  

  i 
  

  

  ■ 
  

  

  

  

  (1 
  +sin 
  2 
  1 
  tar 
  

  

  3 
  0)2 
  

  

  (sec 
  2 
  + 
  cot 
  2 
  

  

  o- 
  

  

  hence 
  from 
  (6) 
  

  

  and 
  

  

  (11) 
  

  

  

  

  

  

  os 
  X 
  

  

  h 
  sin 
  2 
  6 
  

  

  — 
  4- 
  

  

  A 
  sin 
  2 
  6 
  

   = 
  ± 
  B 
  ' 
  

  

  

  

  — 
  ZC 
  

  

  1 
  

  

  

  

  

  ' 
  (d" 
  + 
  h 
  2 
  

  

  sin 
  2 
  0)* 
  

  

  

  From 
  these 
  and 
  

  

  (9) 
  

  

  

  

  

  

  cos 
  cp 
  sin 
  a 
  

  

  (A 
  2 
  + 
  W) 
  = 
  

  

  AB 
  cos 
  

  

  ,0±^B 
  sin 
  2 
  0, 
  

  

  

  and 
  hence 
  from 
  

  

  (2) 
  

  

  

  

  

  

  a- 
  s 
  

  

  

  A 
  2 
  + 
  B 
  2 
  

  

  s 
  

  

  A 
  2 
  + 
  6? 
  2 
  + 
  A 
  2 
  sin 
  

  

  

  

  (13) 
  

   Am 
  A 
  cos 
  d^h 
  sin 
  2 
  4m 
  A 
  cos 
  6 
  ± 
  h 
  sin 
  2 
  d 
  K 
  ' 
  

  

  Since 
  m 
  is 
  the 
  known 
  mass 
  of 
  the 
  planet, 
  and 
  0, 
  s 
  and 
  A 
  are 
  

   known 
  functions 
  of 
  ft), 
  equation 
  (13) 
  gives 
  directly 
  the 
  value 
  of 
  

   @, 
  the 
  semi-axis 
  major 
  of 
  the 
  new 
  orbit 
  (£ 
  in 
  terms 
  of 
  d 
  y 
  k 
  

   and 
  ft). 
  

  

  14. 
  For 
  a 
  particular 
  case 
  of 
  approach, 
  equation 
  (13) 
  is 
  con- 
  

   venient 
  for 
  computation. 
  We 
  may, 
  however, 
  now 
  treat 
  d, 
  h 
  and 
  

   ft) 
  as 
  independent 
  variables 
  whose 
  varying 
  values 
  may 
  express 
  

   all 
  the 
  different 
  possible 
  cases 
  of 
  approach 
  of 
  the 
  comet 
  to 
  the 
  

   planet 
  so 
  far 
  as 
  change 
  of 
  periodic 
  time 
  of 
  the 
  comet 
  is 
  con- 
  

   cerned. 
  The 
  dependence 
  of 
  @ 
  upon 
  the 
  three 
  variables 
  cannot 
  

   be 
  very 
  easily 
  represented 
  graphically 
  in 
  a 
  single 
  plane 
  dia- 
  

   gram. 
  But 
  by 
  giving 
  to 
  ft) 
  successive 
  values 
  in 
  multiples 
  of 
  

   10°, 
  viz 
  : 
  fi>=10°, 
  20°, 
  30°, 
  etc., 
  to 
  170°, 
  I 
  have 
  prepared 
  a 
  

   series 
  of 
  diagrams 
  to 
  exhibit 
  in 
  each 
  case 
  in 
  succession 
  the 
  

   relation 
  of 
  @ 
  to 
  the 
  other 
  two 
  variables. 
  The 
  values 
  of 
  #, 
  s 
  

   and 
  A 
  for 
  the 
  several 
  values 
  of 
  co 
  were 
  needed 
  in 
  making 
  the 
  

   diagrams 
  and 
  they 
  are 
  given 
  in 
  Table 
  I. 
  Equations 
  (4), 
  (5) 
  

  

  