﻿Energetics 
  of 
  undisturbed 
  Planetary 
  Motion. 
  185 
  

  

  and 
  the 
  equation 
  of 
  motion 
  is 
  

  

  v 
  2 
  (v 
  2 
  + 
  V 
  i 
  COS 
  6) 
  = 
  — 
  , 
  

  

  or 
  fjb 
  

  

  1 
  + 
  tf 
  COS 
  

  

  6> 
  W 
  

  

  the 
  equation 
  again 
  of 
  the 
  path. 
  But 
  we 
  have 
  already 
  

   obtained 
  by 
  considerations 
  of 
  the 
  angular 
  momentum 
  the 
  

   equation 
  

  

  h 
  

   l 
  + 
  ecos<9' 
  {i} 
  

  

  and 
  therefore 
  we 
  obtain 
  the 
  very 
  simple 
  and 
  remarkable 
  

   relations 
  

  

  * 
  = 
  £ 
  = 
  «, 
  ....... 
  (8) 
  

  

  V 
  2 
  Vx 
  

  

  that 
  is 
  the 
  double 
  rate 
  of 
  description 
  of 
  area 
  by 
  the 
  radius 
  

   vector 
  is 
  equal 
  to 
  the 
  ratio 
  of 
  the 
  " 
  intensity 
  of 
  the 
  centre 
  " 
  

   to 
  the 
  constant 
  component 
  of 
  velocity 
  at 
  right 
  angles 
  to 
  the 
  

   radius 
  vector. 
  

  

  The 
  time 
  t 
  occupied 
  in 
  describing 
  any 
  part 
  of 
  an 
  orbit 
  is 
  

   thus 
  equal 
  to 
  2Av 
  2 
  /jjl, 
  where 
  A 
  is 
  the 
  corresponding 
  area 
  

   swept 
  over 
  by 
  the 
  radius 
  vector. 
  This, 
  expressed 
  in 
  terms 
  

   of 
  the 
  focal 
  radii 
  and 
  the 
  chord 
  of 
  the 
  arc 
  described 
  and 
  the 
  

   axes, 
  is 
  Lambert's 
  theorem. 
  

  

  A 
  steamer 
  rounding 
  a 
  buoy, 
  in 
  a 
  uniform 
  tidal 
  stream, 
  

   with 
  constant 
  speed 
  v 
  2 
  always 
  directed 
  at 
  right 
  angles 
  to 
  a 
  

   line 
  joining 
  it 
  with 
  the 
  buoy, 
  describes 
  a 
  conic 
  section 
  with 
  

   reference 
  to 
  the 
  land. 
  The 
  curve 
  is 
  a 
  hyperbola 
  if 
  v 
  2 
  is 
  less 
  

   than 
  the 
  speed 
  vi 
  of 
  the 
  stream, 
  and 
  an 
  ellipse 
  in 
  the 
  contrary 
  

   case. 
  The 
  focal 
  axis 
  is 
  at 
  right 
  angles 
  to 
  the 
  stream, 
  and 
  

   e 
  = 
  v 
  } 
  lv 
  2 
  . 
  [I 
  believe 
  that 
  this 
  illustration 
  is 
  originally 
  due 
  

   to 
  Greenhill.] 
  

  

  When 
  = 
  0, 
  (7) 
  becomes 
  

  

  a(l-e)(l+e) 
  = 
  h/v 
  2 
  , 
  

   or 
  (with 
  vijv 
  2 
  = 
  e) 
  

  

  

  . 
  . 
  . 
  (9) 
  

  

  