﻿Energetics 
  of 
  undisturbed 
  Planetary 
  Motion. 
  189 
  

  

  difficulty 
  in 
  framing 
  a, 
  formal 
  proof, 
  but 
  the 
  following 
  may- 
  

   be 
  sufficient. 
  

  

  At 
  a 
  considerable 
  distance 
  from 
  the 
  centre 
  of 
  force 
  the 
  

   kinetic 
  energy 
  in 
  the 
  hyperbolic 
  orbit 
  will 
  have 
  become 
  

   practically 
  constant 
  and 
  equal 
  to 
  fjuj2a. 
  For 
  r 
  has 
  become 
  

   very 
  great 
  and 
  fi/r 
  very 
  small, 
  so 
  that 
  ^v 
  2 
  = 
  fi/2a. 
  The 
  

   planet 
  therefore 
  moves 
  along 
  the 
  curve, 
  ultimately 
  along 
  

   the 
  asymptote, 
  with 
  more 
  and 
  more 
  exactly 
  constant 
  speed, 
  

   (fi/a)*, 
  and 
  as 
  it 
  continues 
  at 
  this 
  speed, 
  in 
  its 
  coming 
  and 
  its 
  

   going, 
  for 
  an 
  infinite 
  time, 
  the 
  time-average 
  of 
  the 
  kinetic 
  

   energy 
  is 
  fi/2a, 
  as 
  in 
  the 
  elliptic 
  orbit. 
  

  

  It 
  may 
  seem 
  a 
  hard 
  saying 
  that 
  the 
  time-average 
  of 
  the 
  

   kinetic 
  energy 
  in 
  a 
  parabolic 
  orbit 
  is 
  zero, 
  but 
  in 
  this 
  case 
  

   the 
  energy 
  equation 
  is 
  \v 
  2 
  ~ix\r, 
  and 
  so 
  v 
  is 
  very 
  small 
  when 
  

   r 
  is 
  very 
  large. 
  Thus 
  during 
  an 
  infinite 
  time 
  the 
  value 
  of 
  

   j;o 
  2 
  is 
  the 
  evanescent 
  quantity 
  fju/r, 
  and 
  thus 
  the 
  time-average 
  

   of 
  the 
  kinetic 
  energy 
  is 
  evanescent. 
  

  

  This 
  theory 
  leads 
  to 
  the 
  result 
  that 
  the 
  exhaustion 
  of 
  

   gravitational 
  potential 
  energy 
  alone 
  cannot 
  have 
  led 
  to 
  

   motion 
  of 
  a 
  planet 
  or 
  comet 
  in 
  a 
  hyperbolic 
  orbit. 
  It 
  would 
  

   seem 
  that 
  the 
  necessary 
  excess 
  of 
  mean 
  kinetic 
  energy 
  must 
  

   have 
  been 
  produced 
  by 
  some 
  cataclysm 
  within 
  a 
  body, 
  from 
  

   which 
  the 
  planet 
  was 
  thrown 
  off 
  after 
  the 
  sun 
  had 
  done 
  the 
  

   work 
  of 
  bringing 
  the 
  body 
  within 
  a 
  finite 
  distance 
  r 
  of 
  the 
  

   centre 
  of 
  force. 
  Only 
  by 
  exhaustion 
  of 
  internal 
  energy 
  

   does 
  it 
  seem 
  possible 
  to 
  make 
  up 
  the 
  necessary 
  additional 
  

   energy. 
  

  

  The 
  question 
  of 
  equipartition 
  of 
  energy 
  between 
  the 
  

   different 
  stars 
  has 
  been 
  a 
  good 
  deal 
  discussed. 
  If 
  a 
  system 
  

   of 
  stars 
  has 
  its 
  origin 
  in 
  the 
  exhaustion 
  of 
  potential 
  energy 
  

   by 
  the 
  attraction 
  of 
  some 
  great 
  central 
  system, 
  the 
  energy 
  

   relations 
  here 
  discussed 
  would 
  appear 
  to 
  negative 
  equipartition. 
  

   If 
  there 
  is 
  an 
  approach 
  to 
  equipartition 
  of 
  kinetic 
  energy 
  of 
  

   translational 
  motion, 
  and 
  there 
  is 
  evidence 
  apparently 
  that 
  

   the 
  more 
  massive 
  stars 
  move 
  the 
  slower, 
  such 
  an 
  origin 
  

   becomes 
  on 
  one 
  more 
  ground 
  improbable. 
  

  

  The 
  University, 
  Glasgow, 
  

   Dec. 
  31, 
  1917. 
  

  

  Phil. 
  Mag. 
  S. 
  6. 
  Vol. 
  35. 
  No. 
  206. 
  Feb. 
  1918. 
  

  

  