﻿486 
  H. 
  A. 
  Newton 
  — 
  Capture 
  of 
  Comets 
  ~by 
  Planets, 
  

  

  If 
  in 
  a 
  diagram 
  constructed 
  for 
  to 
  = 
  BAH 
  the 
  isergonal 
  curve 
  

   be 
  drawn 
  for 
  @ 
  = 
  r, 
  those 
  comets 
  for 
  which 
  d 
  and 
  A 
  represent 
  

   points 
  within 
  that 
  isergonal 
  curve 
  will 
  after 
  perturbation 
  have 
  

   velocities 
  represented 
  by 
  lines 
  drawn 
  from 
  points 
  in 
  ET 
  to 
  A, 
  

   while 
  comets 
  for 
  which 
  d 
  and 
  A 
  represent 
  points 
  outside 
  that 
  

   isergonal 
  curve 
  will 
  after 
  perturbation 
  have 
  directions 
  of 
  

   motion 
  represented 
  by 
  lines 
  drawn 
  to 
  A 
  from 
  points 
  in 
  EHK. 
  

  

  Fig. 
  19. 
  

  

  The 
  number 
  of 
  comets 
  having 
  motions 
  represented 
  by 
  lines 
  to 
  

   A 
  from 
  points 
  in 
  ET 
  will 
  be 
  proportional 
  to 
  the 
  area 
  of 
  the 
  

   isergonal 
  curve 
  @ 
  = 
  r. 
  Let 
  the 
  angle 
  BAS 
  represent 
  a 
  limit- 
  

   ing 
  value 
  to" 
  of 
  distance 
  of 
  quits 
  of 
  comets 
  from 
  Jupiter's 
  quit 
  

   after 
  perturbation. 
  The 
  comets 
  which 
  are 
  thus 
  limited 
  and 
  at 
  

   the 
  same 
  time 
  have 
  @<^ 
  will 
  be 
  moving 
  in 
  lines 
  directed 
  to 
  

   A 
  from 
  points 
  in 
  the 
  area 
  bounded 
  by 
  the 
  straight 
  lines 
  SA 
  

   and 
  AF, 
  and 
  the 
  arcs 
  FD 
  and 
  DS. 
  Let 
  to 
  receive 
  an 
  incre- 
  

   ment 
  dto 
  — 
  HA 
  and 
  let 
  a 
  new 
  semicircu 
  inference 
  be 
  drawn 
  

   with 
  BA 
  as 
  radius. 
  To 
  the 
  elemental 
  arc 
  HA 
  will 
  correspond 
  

   the 
  elemental 
  area 
  along 
  the 
  semicircumference 
  KET. 
  If: 
  ET 
  

   lies 
  wholly 
  in 
  SAFD 
  the 
  number 
  of 
  comets 
  that 
  pass 
  the 
  

   planet 
  in 
  a 
  unit 
  of 
  time 
  having 
  initial 
  angles 
  of 
  direction 
  with 
  

   Jupiter's 
  motion 
  between 
  to 
  and 
  to+dto 
  will 
  be 
  equal 
  to 
  the 
  

   area 
  of 
  the 
  isergonal 
  curve 
  for 
  % 
  — 
  r 
  multiplied 
  by 
  the 
  elemen- 
  

   tal 
  number 
  \n 
  sin 
  tod 
  to, 
  and 
  by 
  the 
  relative 
  velocity 
  v 
  sin 
  6 
  

   of 
  the 
  comet 
  perpendicular 
  to 
  the 
  isergonal 
  area. 
  If 
  the 
  area 
  

   of 
  the 
  isergonal 
  curve 
  be 
  represented 
  by 
  #r 
  sin 
  6, 
  then 
  this 
  

   product 
  will 
  be 
  

  

  tf» 
  . 
  n 
  sin 
  GodoD 
  nv 
  ^ 
  7 
  

   . 
  « 
  sin 
  o 
  . 
  = 
  — 
  - 
  (Pels. 
  

  

  s 
  2 
  sin0* 
  ° 
  2 
  4 
  

  

  since 
  \Z%v 
  = 
  sv, 
  and 
  x/2 
  sin 
  tod 
  to 
  = 
  sds. 
  

  

  38. 
  This 
  expresses 
  the 
  elemental 
  number 
  of 
  comets 
  corres- 
  

   ponding 
  to 
  the 
  elemental 
  area 
  Te. 
  The 
  integral 
  of 
  this 
  

   expression, 
  that 
  is, 
  \nvf<i>ds, 
  so 
  taken 
  as 
  to 
  cover 
  the 
  area 
  

  

  