﻿H. 
  A. 
  Newton— 
  Capture 
  of 
  Comets 
  by 
  Planets. 
  198 
  

  

  point 
  of 
  intersection 
  at 
  the 
  instant 
  when 
  the 
  planet's 
  distmice 
  

   therefrom 
  is 
  equal 
  to 
  the 
  planet's 
  distance 
  from 
  the 
  sun 
  multi- 
  

   plied 
  by 
  the 
  ratio 
  of 
  the 
  mass 
  of 
  the 
  planet 
  to 
  the 
  mass 
  of 
  the 
  

  

  ^The 
  relative 
  velocity 
  of 
  the 
  comet 
  on 
  leaving 
  the 
  planet's 
  

   sphere 
  of 
  action 
  would 
  be 
  equal 
  to 
  and 
  directly 
  opposite 
  to 
  the 
  

   planet's 
  velocity 
  («=1), 
  and 
  the 
  comet 
  would 
  be 
  left 
  entirely 
  

   at 
  rest 
  to 
  fall 
  to 
  the 
  sun. 
  This 
  case 
  could 
  not 
  happen 
  for 
  

   planets 
  like 
  the 
  earth 
  where 
  mr 
  is 
  less 
  than 
  the 
  semi-diameter 
  

   of 
  the 
  planet. 
  In 
  the 
  case 
  of 
  the 
  earth 
  mr 
  is 
  less 
  than 
  300 
  

   miles, 
  and 
  actual 
  collision 
  would 
  result. 
  But 
  for 
  Jupiter 
  mr 
  

   is 
  greater 
  than 
  the 
  distance 
  of 
  the 
  second 
  satellite 
  from 
  the 
  

   planet 
  The 
  nearest 
  approach 
  of 
  the 
  comet 
  to 
  the 
  planet 
  

   would 
  be 
  mr 
  (n/2-1) 
  which 
  is 
  more 
  than 
  four 
  times 
  the 
  radius 
  

   of 
  Jupiter. 
  Hence 
  this 
  case 
  of 
  maximum 
  diminution 
  of 
  major 
  

   axis 
  could 
  occur 
  near 
  Jupiter. 
  

  

  Fig. 
  2; 
  w=10' 
  

  

  Fig. 
  3; 
  «=170°. 
  

  

  22. 
  Isergonal 
  ellipse 
  for 
  © 
  = 
  10°.— 
  If 
  we 
  make 
  ©=10 
  the 
  

   vanishing 
  points 
  of 
  the 
  isergonal 
  ellipses 
  will 
  be 
  (Table 
  11) 
  at 
  

  

  d=0 
  aJ-01250, 
  and 
  <*=°> 
  A= 
  - 
  ' 
  15174 
  - 
  In 
  fi 
  S* 
  2 
  let 
  ° 
  E 
  ? 
  nd 
  

   OH 
  be 
  the 
  axes 
  of 
  d 
  and 
  h 
  respectively. 
  The 
  vanishing 
  points 
  

   will 
  be 
  on 
  the 
  axis 
  OH 
  at 
  distances 
  h' 
  and 
  h" 
  above 
  and 
  below 
  

   O 
  Upon 
  this 
  diagram 
  are 
  shown 
  the 
  halves 
  of 
  four 
  isergonal 
  

   ellipses. 
  The 
  scales 
  used 
  for 
  d 
  and 
  h 
  are 
  not 
  equal 
  to 
  each 
  

   other, 
  since 
  the 
  use 
  of 
  the 
  same 
  scale 
  for 
  both 
  coordinates 
  

   would 
  make 
  the 
  figures 
  of 
  inconvenient 
  shape. 
  In 
  this, 
  and 
  

   in 
  all 
  the 
  figures 
  2-18, 
  the 
  unit 
  in 
  d 
  is 
  to 
  the 
  unit 
  in 
  h, 
  as 
  1 
  

   to 
  sin 
  a>. 
  But 
  to 
  indicate 
  more 
  clearly 
  this 
  scale, 
  and 
  at 
  the 
  

   same 
  time 
  to 
  give 
  a 
  kind 
  of 
  shading 
  to 
  a 
  part 
  of 
  the 
  area, 
  there 
  

   are 
  drawn 
  above 
  the 
  radical 
  axis 
  ae 
  lines 
  parallel 
  to 
  O-b, 
  and 
  

   parallel 
  to 
  OH, 
  at 
  intervals 
  of 
  '01 
  ; 
  that 
  is, 
  the 
  sides 
  of 
  each 
  ot 
  

   the 
  small 
  rectangles 
  in 
  the 
  quadrant 
  HOE 
  are 
  -01, 
  or 
  about 
  

  

  