﻿Energeiws 
  of 
  undisturbed 
  Planetary 
  Motion. 
  187 
  

  

  Some 
  years 
  ago 
  I 
  came 
  upon 
  the 
  examination 
  question 
  : 
  

   Prove 
  that 
  the 
  action 
  for 
  a 
  complete 
  revolution 
  of 
  a 
  planet 
  in 
  

   its 
  orbit 
  is 
  independent 
  of 
  the 
  eccentricity 
  of 
  the 
  orbit. 
  The 
  

   following 
  proof 
  of 
  this 
  interesting 
  proposition 
  presented 
  

   itself 
  : 
  it 
  is 
  simple 
  and 
  I 
  think 
  elegant. 
  The 
  action, 
  A, 
  is 
  

   given 
  by 
  the 
  equation 
  

  

  A 
  = 
  §v*dt=$vds, 
  (14) 
  

  

  where 
  the 
  space 
  integration 
  is 
  taken 
  round 
  the 
  orbit 
  and 
  the 
  

   time 
  integration 
  for 
  a 
  complete 
  period. 
  Let 
  p, 
  p' 
  be 
  the 
  

   lengths 
  of 
  the 
  perpendiculars 
  let 
  fall 
  from 
  S 
  and 
  the 
  empty- 
  

   focus 
  on 
  the 
  tangent 
  at 
  P 
  to 
  the 
  orbit. 
  We 
  have 
  

  

  pv 
  = 
  7i 
  = 
  fi/v 
  2) 
  or 
  v 
  = 
  (jilvj?)p* 
  = 
  (p,/ab 
  2 
  )2p', 
  by 
  (9), 
  

  

  Hence 
  

  

  A=j,& 
  = 
  (j 
  2 
  ) 
  i 
  jy^ 
  = 
  2,r( 
  /i 
  a)i, 
  . 
  (15) 
  

  

  since 
  \ 
  p'ds 
  is 
  2wab, 
  twice 
  the 
  area 
  of 
  the 
  (elliptic) 
  orbit. 
  

   The 
  action 
  is 
  thus 
  independent 
  of 
  the 
  eccentricity, 
  and 
  

   there 
  is 
  no 
  variation 
  of 
  the 
  total 
  action 
  from 
  one 
  orbit 
  

   to 
  another, 
  provided 
  both 
  possess 
  the 
  same 
  major 
  axis. 
  

   The 
  action 
  may 
  also 
  be 
  written 
  as 
  27rbv 
  2 
  . 
  [There 
  is 
  no 
  

   difficulty 
  in 
  writing 
  down 
  an 
  expression 
  for 
  the 
  action 
  in 
  

   any 
  finite 
  part 
  of 
  the 
  orbit.] 
  

  

  If 
  T 
  be 
  the 
  period 
  we 
  get 
  by 
  (13) 
  

  

  A 
  = 
  T^=2T£- 
  (16) 
  

  

  a 
  2a 
  K 
  

  

  This 
  is 
  the 
  time 
  integral 
  of 
  v 
  2 
  . 
  The 
  time 
  average 
  of 
  the 
  

   kinetic 
  energy 
  in 
  the 
  orbit 
  is 
  fi/2a. 
  

  

  From 
  this 
  proof 
  * 
  it 
  appeared 
  that 
  the 
  action 
  is 
  pro- 
  

   portional 
  to 
  the 
  area 
  swept 
  over 
  in 
  any 
  time 
  by 
  the 
  radius 
  

   vector 
  from 
  the 
  empty 
  focus 
  to 
  the 
  planet. 
  Thus 
  while 
  the 
  

   radius 
  vector 
  from 
  the 
  sun 
  to 
  the 
  planet 
  is 
  the 
  timekeeper, 
  

   measuring 
  as 
  it 
  does 
  time 
  in 
  the 
  orbit 
  by 
  the 
  area 
  it 
  sweeps 
  

   over, 
  the 
  other 
  radius 
  vector 
  l6 
  keeps 
  " 
  the 
  action. 
  This 
  

   proposition 
  I 
  found 
  had 
  already 
  been 
  stated 
  by 
  Professor 
  

   Tait. 
  

  

  * 
  Another 
  proof 
  naturally 
  occurs 
  in 
  which 
  the 
  integration 
  is 
  effected 
  

   with 
  the 
  aid 
  of 
  the 
  eccentric 
  angle 
  : 
  but 
  it 
  is 
  long 
  and 
  uusuggestive. 
  

   Since 
  this 
  paper 
  was 
  sent 
  in 
  I 
  have 
  found 
  a 
  memoir 
  by 
  Grinwis, 
  Akad. 
  

   van 
  Wetens., 
  Amsterdam, 
  ix. 
  1891-2, 
  in 
  which 
  the 
  proposition 
  regarding 
  

   the 
  action 
  is 
  given. 
  Probably 
  this 
  was 
  the 
  origin 
  of 
  the 
  examination 
  

   question. 
  

  

  