﻿188 
  Prof. 
  A. 
  Gray 
  on 
  Hodographic 
  Treatment 
  and 
  

  

  9. 
  The 
  term 
  fi/r 
  in 
  the 
  energy 
  equation 
  is 
  the 
  potential 
  energy 
  

   exhausted 
  when 
  the 
  planet 
  is 
  brought 
  by 
  the 
  sun's 
  attraction 
  

   from 
  infinity 
  to 
  the 
  distance 
  r. 
  Thus 
  we 
  obtain 
  by 
  (11) 
  the 
  

   curious 
  theorem 
  that 
  the 
  time-average 
  value 
  of 
  this 
  term 
  for 
  a 
  

   complete 
  revolution 
  of 
  the 
  planet 
  is 
  twice 
  the 
  time-average 
  

   of 
  the 
  kinetic 
  energy 
  in 
  the 
  orbit, 
  that 
  in 
  fact 
  the 
  time- 
  

   average 
  of 
  this 
  exhaustion 
  of 
  potential 
  energy 
  is 
  equal 
  to 
  

   the 
  action. 
  This 
  is 
  also, 
  of 
  course, 
  independent 
  of 
  the 
  

   eccentricity. 
  

  

  As 
  a 
  particular 
  case 
  of 
  this 
  the 
  kinetic 
  energy 
  of 
  a 
  planet 
  

   in 
  a 
  circular 
  orbit 
  is 
  half: 
  the 
  potential 
  energy 
  exhausted 
  in 
  

   the 
  journey 
  from 
  infinity. 
  Hence 
  also, 
  if 
  the 
  planet 
  were 
  

   transferred 
  from 
  one 
  circular 
  orbit 
  to 
  one 
  of 
  (say) 
  smaller 
  

   radius, 
  the 
  increase 
  of 
  kinetic 
  energy 
  would 
  be 
  only 
  one 
  

   half 
  of 
  the 
  additional 
  potential 
  energy 
  exhausted 
  in 
  the 
  

   passage. 
  

  

  This 
  result 
  for 
  the 
  circular 
  orbit 
  is 
  of 
  course 
  well 
  known 
  ; 
  

   the 
  corresponding 
  relation 
  which 
  holds 
  for 
  the 
  mean 
  kinetic 
  

   energy 
  in 
  an 
  elliptic 
  orbit 
  was, 
  I 
  believe, 
  first 
  stated 
  by 
  myself 
  

   in 
  a 
  letter 
  in 
  'Nature,' 
  August 
  7, 
  1913. 
  The 
  fixed 
  ratio 
  (J) 
  

   of 
  the 
  mean 
  orbital 
  kinetic 
  energy 
  to 
  the 
  mean 
  potential 
  

   energy, 
  exhausted 
  to 
  the 
  different 
  points 
  of 
  the 
  orbit, 
  is 
  

   curious 
  ; 
  but 
  there 
  is 
  always 
  a 
  fixed 
  ratio 
  of 
  the 
  energy 
  

   dissipated 
  in 
  the 
  interactions 
  of 
  bodies 
  to 
  the 
  whole 
  available 
  

   energy. 
  For 
  example, 
  a 
  body 
  of 
  mass 
  m 
  moving 
  with 
  

   speed 
  v 
  collides 
  inelastically 
  with 
  a 
  body 
  of 
  mass 
  mf 
  at 
  rest 
  ; 
  

   and 
  the 
  kinetic 
  energy 
  dissipated 
  bears 
  to 
  the 
  original 
  kinetic 
  

   energy 
  the 
  ratio 
  m'/(m 
  + 
  m), 
  which 
  is 
  quite 
  independent 
  of 
  

   the 
  details 
  of 
  the 
  action 
  between 
  the 
  bodies. 
  

  

  The 
  theorem 
  for 
  passage 
  from 
  one 
  elliptic 
  orbit 
  to 
  another 
  

   is 
  exactly 
  parallel 
  to 
  that 
  stated 
  above 
  for 
  a 
  circular 
  orbit. 
  

  

  In 
  the 
  planetary 
  case 
  then, 
  the 
  bodies 
  are 
  only 
  left 
  moving 
  

   in 
  elliptic 
  orbits, 
  when 
  the 
  proper 
  adjustment 
  has 
  taken 
  place; 
  

   others 
  if 
  left 
  moving 
  too 
  slowly 
  will 
  fall 
  into 
  the 
  central 
  

   body, 
  or 
  if 
  moving 
  too 
  quickly 
  may 
  recede 
  from 
  the 
  central 
  

   body 
  to 
  undergo 
  further 
  energy 
  modifications 
  by 
  collision 
  or 
  

   otherwise. 
  

  

  Now 
  let 
  us 
  compare 
  a 
  hyperbolic 
  orbit 
  with 
  an 
  elliptic 
  orbit 
  

   as 
  regards 
  this 
  affair 
  of 
  energy. 
  We 
  have 
  the 
  theorem 
  that 
  

   the 
  kinetic 
  energy 
  at 
  distance 
  r 
  in 
  a 
  hyperbolic 
  orbit 
  exceeds, 
  

   and 
  in 
  an 
  elliptic 
  orbit 
  falls 
  short 
  of, 
  the 
  potential 
  energy 
  

   exhausted 
  from 
  infinity 
  to 
  this 
  distance 
  by 
  the 
  mean 
  value 
  of 
  

   the 
  kinetic 
  energy 
  in 
  the 
  orbit. 
  This 
  result 
  has 
  not 
  been 
  

   formally 
  proved 
  for 
  the 
  hyperbolic 
  orbit, 
  but 
  has 
  been 
  inferred 
  

   by 
  analogy 
  from 
  the 
  result 
  for 
  the 
  ellipse. 
  There 
  is 
  no 
  

  

  