﻿484 
  H. 
  A. 
  Newton 
  — 
  Capture 
  of 
  Comets 
  by 
  Planets. 
  

  

  comets 
  whose 
  quits 
  are 
  in 
  the 
  element 
  dS 
  of 
  the 
  celestial 
  sphere 
  

   and 
  that 
  will 
  pass 
  the 
  planet 
  in 
  a 
  unit 
  of 
  time 
  in 
  such 
  a 
  way 
  

   as 
  to 
  have 
  their 
  periodic 
  times 
  reduced 
  below 
  a 
  given 
  period 
  

   will 
  be 
  equal 
  to 
  the 
  area 
  inclosed 
  in 
  the 
  corresponding 
  isergo- 
  

   nal 
  curve 
  multiplied 
  by 
  the 
  velocity 
  perpendicular 
  to 
  the 
  

  

  plane, 
  v 
  sin 
  6, 
  and 
  by 
  the 
  factor 
  -— 
  -. 
  If 
  @ 
  is 
  the 
  semi-major 
  

  

  axis 
  of 
  the 
  orbit 
  for 
  the 
  limiting 
  periodic 
  time, 
  the 
  area 
  of 
  the 
  

   corresponding 
  isergonal 
  curve 
  will 
  be 
  (Art. 
  17). 
  

  

  n 
  /4:m 
  2 
  (df 
  /2m@ 
  cos 
  6 
  mrX 
  2 
  \ 
  

   im0\ 
  7 
  \ 
  s~~ 
  ~~ 
  Vj 
  / 
  

  

  For 
  dS 
  we 
  may, 
  as 
  before, 
  take 
  2tt 
  sin 
  co 
  dco, 
  and 
  we 
  shall 
  then 
  

  

  have 
  

  

  nn 
  /» 
  . 
  f4m 
  8 
  @ 
  2 
  /2m@cos0 
  mr\-~\ 
  _ 
  

   N 
  = 
  -Jv 
  smoo\-^- 
  - 
  ( 
  7 
  ) 
  \doo. 
  

  

  The 
  integration 
  must 
  extend 
  through 
  the 
  positive 
  values 
  of 
  

   the 
  quantity 
  in 
  square 
  brackets 
  beginning 
  at 
  co 
  — 
  0. 
  [In 
  case 
  

   a> 
  — 
  gives 
  a 
  negative 
  value 
  for 
  the 
  quantity 
  in 
  square 
  brack- 
  

   ets 
  we 
  must 
  integrate 
  between 
  the 
  two 
  values 
  of 
  co 
  correspond- 
  

   ing 
  to 
  the 
  zero 
  value 
  of 
  the 
  bracketed 
  quantity.] 
  We 
  may 
  

   make 
  S 
  the 
  independent 
  variable 
  by 
  the 
  equations 
  

   sds 
  — 
  v/2 
  sin 
  co 
  dco, 
  v 
  ox 
  /2 
  = 
  sv, 
  and 
  2s 
  cos 
  6 
  = 
  1— 
  s 
  2 
  . 
  

  

  These 
  give 
  : 
  

  

  N 
  = 
  l« 
  n 
  m>vf[ 
  m 
  <-( 
  ®- 
  r 
  -® 
  ?f]ds. 
  

  

  33. 
  If 
  now 
  we 
  require 
  the 
  number 
  of 
  comets 
  which 
  in 
  each 
  

   unit 
  of 
  time 
  shall 
  pass 
  the 
  planet 
  in 
  such 
  way 
  as 
  that 
  they 
  

   shall 
  have 
  after 
  the 
  passage 
  respectively 
  less 
  than 
  one-half, 
  

   once, 
  three-halves, 
  and 
  twice, 
  the 
  planet's 
  period 
  of 
  revolution, 
  

  

  we 
  may 
  place 
  @ 
  = 
  rT^, 
  and 
  make 
  T 
  equal 
  successively 
  to 
  \, 
  

   1, 
  f, 
  and 
  2, 
  and 
  compute 
  in 
  each 
  case 
  the 
  value 
  of 
  N" 
  as 
  given 
  

   in 
  the 
  last 
  article. 
  The 
  results 
  are 
  found 
  to 
  be 
  irnm'r^v 
  mul- 
  

   tiplied 
  severally 
  by 
  the 
  coefficients 
  0*139, 
  0*925, 
  1*875, 
  and 
  

   2*943. 
  

  

  34. 
  By 
  comparing 
  the 
  results 
  of 
  Arts. 
  27 
  and 
  33. 
  and 
  mak- 
  

   ing 
  the 
  assumptions 
  of 
  Art. 
  26, 
  we 
  have 
  the 
  proposition, 
  that 
  

   the 
  number 
  of 
  comets 
  which 
  in 
  a 
  given 
  period 
  of 
  time 
  pass 
  

   their 
  perihelia 
  nearer 
  to 
  the 
  sun 
  than 
  a 
  given 
  planet, 
  is 
  to 
  the 
  

   number 
  of 
  comets 
  whose 
  periodic 
  times 
  are 
  reduced 
  by 
  the 
  per- 
  

   turbing 
  action 
  of 
  the 
  planet 
  so 
  as 
  to 
  be 
  less 
  severally 
  than 
  one- 
  

   half, 
  once, 
  three 
  halves, 
  and 
  twice, 
  the 
  periodic 
  time 
  of 
  the 
  

   planet, 
  as 
  itnity 
  is 
  to 
  the 
  square 
  of 
  the 
  mass 
  of 
  the 
  planet 
  mul- 
  

   tiplied 
  severally 
  by 
  0*139, 
  0*925, 
  1*876 
  and 
  2*943. 
  

  

  