﻿H. 
  A. 
  Newton 
  — 
  Capture 
  of 
  Comets 
  by 
  Planets. 
  487 
  

  

  AFDS 
  Trill 
  give 
  the 
  number 
  of 
  comets 
  which 
  in 
  a 
  unit 
  of 
  time 
  

   will 
  pass 
  the 
  planet 
  in 
  such 
  a 
  way 
  as 
  to 
  have 
  @<^r 
  and 
  

   a/<BAS. 
  ^Vhen 
  the 
  elemental 
  area 
  does 
  not 
  extend 
  from 
  

   the 
  arc 
  DS 
  to 
  the 
  line 
  BA, 
  the 
  area 
  of 
  another 
  appropriate 
  

   isergonal 
  curve 
  is 
  to 
  be 
  used 
  in 
  determining 
  0. 
  

   By 
  Art. 
  17 
  we 
  have 
  

  

  A 
  * 
  /@ 
  — 
  >'— 
  ©AH 
  

  

  For 
  the 
  elemental 
  areas 
  of 
  the 
  surface 
  AFDS 
  which 
  end 
  on 
  

   the 
  arc 
  DS 
  we 
  make 
  @ 
  = 
  r, 
  and 
  let 
  <£ 
  be 
  the 
  resulting 
  value 
  

   of 
  <t> 
  ; 
  then 
  <z> 
  = 
  7r^V(4-s- 
  2 
  ). 
  

  

  For 
  elemental 
  areas 
  that 
  end 
  on 
  the 
  radius 
  AS 
  the 
  values 
  of 
  

   @ 
  on 
  that 
  line 
  are 
  functions 
  of 
  s. 
  To 
  compute 
  them 
  let 
  v' 
  be 
  

   the 
  comet's 
  velocity 
  in 
  its 
  orbit 
  about 
  the 
  sun, 
  and 
  hence 
  equal 
  

   to 
  the 
  distance 
  of 
  the 
  point 
  on 
  AS 
  from 
  A 
  ; 
  then, 
  by 
  the 
  tri- 
  

   angle 
  of 
  velocities 
  

  

  v*+v 
  f 
  *-^2v'v 
  j 
  cos 
  0D 
  f/ 
  = 
  v 
  Q 
  2 
  =s 
  2 
  v 
  2 
  . 
  

  

  Again 
  by 
  the 
  laws 
  of 
  gravitation, 
  

  

  v' 
  2 
  = 
  (2-^\v 
  2 
  

   \ 
  @7 
  ' 
  

  

  V 
  / 
  V 
  

  

  Hence 
  6- 
  2 
  =3 
  24/2 
  cos 
  go", 
  

  

  @_3 
  — 
  s 
  2 
  — 
  2 
  cos 
  2 
  gj"^z2 
  cos 
  oo"(s 
  2 
  — 
  sin 
  2 
  09")* 
  

   ~r~~ 
  9 
  — 
  8 
  cos 
  2 
  oj" 
  — 
  6s 
  2 
  + 
  s 
  i 
  

  

  Let 
  <!>' 
  and 
  <i>" 
  be 
  the 
  two 
  values 
  of 
  t> 
  obtained 
  by 
  substituting 
  

   in 
  these 
  values 
  of 
  @, 
  #" 
  representing 
  the 
  value 
  for 
  the 
  point 
  

   nearer 
  to 
  A. 
  

  

  39. 
  If 
  to" 
  = 
  90°, 
  and 
  therefore 
  cos 
  to" 
  = 
  0, 
  we 
  have 
  along 
  

   the 
  limiting 
  line, 
  the 
  two 
  values 
  of 
  @ 
  equal, 
  hence 
  

  

  — 
  = 
  5, 
  and 
  <P'=- 
  

  

  "3-s 
  21 
  s 
  2 
  (3-s 
  2 
  ) 
  2 
  ' 
  

  

  so 
  that 
  the 
  number 
  of 
  comets 
  having 
  quits 
  less 
  than 
  90° 
  from 
  

   Jupiter's 
  quit 
  and 
  @0 
  is 
  

  

  4/2 
  

  

  -J;^r^ 
  •*=—-- 
  vj^- 
  s)ds 
  -y, 
  tp^j 
  

  

  |/2 
  

  

  7znvm 
  2 
  r 
  

  

  (7 
  + 
  ^2) 
  = 
  '7012 
  7tnvm 
  2 
  r 
  2 
  . 
  

  

  Since 
  the 
  whole 
  number 
  of 
  such 
  comets 
  is 
  (Art, 
  33) 
  equal 
  to 
  

   •925 
  Trnvm'r 
  2 
  ) 
  the 
  number 
  of 
  comets 
  the 
  distance 
  of 
  whose 
  quits 
  

  

  Am. 
  Jour. 
  Sci. 
  — 
  Third 
  Series, 
  Vol. 
  XLII, 
  No. 
  252. 
  — 
  "December, 
  1891. 
  

   33 
  

  

  