﻿194 
  II. 
  A. 
  Newton 
  — 
  Capture 
  of 
  Comets 
  ~by 
  Planets. 
  

  

  925,000 
  miles. 
  Only 
  the 
  positive 
  values 
  of 
  d 
  are 
  represented 
  

   in 
  the 
  figures. 
  The 
  positive 
  vanishing 
  point 
  being 
  1-250 
  of 
  

   these 
  divisions 
  above 
  O, 
  and 
  the 
  negative 
  vanishing 
  point 
  

   15,174 
  below 
  O, 
  we 
  lay 
  off 
  Oa=^{h' 
  + 
  A") 
  = 
  -6*962 
  divisions, 
  

   and 
  draw 
  ae 
  for 
  the 
  radical 
  axis. 
  The 
  smallest 
  positive 
  value 
  

   of 
  @ 
  is 
  (Table 
  II) 
  3*04. 
  As 
  @ 
  increases 
  from 
  3*04 
  the 
  ellipse 
  

   increases 
  in 
  size, 
  and 
  the 
  innermost 
  curve 
  represents 
  what 
  it 
  

   becomes 
  when 
  @=5. 
  The 
  second 
  curve 
  (separating 
  the 
  blank 
  

   and 
  shaded 
  areas) 
  corresponds 
  to 
  @=20. 
  Any 
  parabolic 
  comet 
  

   passing 
  Jupiter 
  with 
  an 
  original 
  angle 
  of 
  o) 
  = 
  10°, 
  and 
  having 
  

   d 
  and 
  h 
  such 
  as 
  to 
  be 
  represented 
  by 
  a 
  point 
  within 
  the 
  blank 
  

   area 
  of 
  1ig. 
  2 
  will 
  leave 
  the 
  vicinity 
  of 
  the 
  planet 
  in 
  an 
  elliptic 
  

   orbit 
  whose 
  semi-axis 
  major 
  is 
  less 
  than 
  20, 
  and 
  whose 
  period 
  

   therefore 
  is 
  less 
  than 
  90 
  years. 
  

  

  The 
  larger 
  curve 
  that 
  lies 
  above 
  ae 
  in 
  the 
  shaded 
  area 
  is 
  the 
  

   isergonal 
  ellipse 
  for 
  @=50. 
  As 
  @ 
  increases 
  the 
  lower 
  part 
  of 
  

   the 
  curve 
  tends 
  to 
  approach 
  the 
  radical 
  axis 
  ae, 
  with 
  which 
  it 
  

   coincides 
  when 
  @=oc 
  . 
  For 
  points 
  in 
  the 
  area 
  below 
  ae 
  (dis- 
  

   tinguished 
  by 
  the 
  oblique-line 
  shading), 
  the 
  planet 
  increases 
  the 
  

   velocity 
  of 
  the 
  comets, 
  and 
  the 
  comet 
  would 
  be 
  thrown 
  per- 
  

   manently 
  out 
  of 
  the 
  solar 
  system. 
  The 
  smallest 
  semi-trans- 
  

   verse 
  axis, 
  the 
  one 
  corresponding 
  to 
  the 
  vanishing 
  ellipse 
  is 
  

   (Table 
  II) 
  36'90, 
  and 
  the 
  isergonal 
  curve 
  for 
  @= 
  — 
  50 
  is 
  drawn 
  

   in 
  the 
  figure. 
  

  

  