﻿Energetics 
  of 
  undisturbed 
  Planetary 
  Motion. 
  183 
  

  

  be 
  the 
  angular 
  speed 
  with 
  which 
  SP 
  is 
  turning 
  about 
  S, 
  and 
  

   r 
  = 
  SP, 
  we 
  have 
  r 
  2 
  6 
  = 
  h, 
  where 
  h 
  is 
  a 
  constant. 
  Thus 
  6 
  = 
  h/r 
  2 
  . 
  

   From 
  a 
  centre 
  C 
  let 
  a 
  circle 
  be 
  described 
  with 
  some 
  radius 
  2a 
  

   in 
  the 
  plane 
  of 
  the 
  path 
  or 
  in 
  a 
  parallel 
  plane. 
  Let 
  a 
  radius 
  

   CH 
  of 
  the 
  circle 
  be 
  always 
  parallel 
  to 
  SP. 
  The 
  velocity 
  of 
  H 
  

   is 
  2a0, 
  and 
  is 
  therefore 
  proportional 
  to 
  1/r 
  2 
  . 
  If 
  the 
  force 
  per 
  

   unit 
  mass 
  on 
  the 
  planet 
  at 
  distance 
  r 
  be 
  fi/r 
  2 
  , 
  the 
  acceleration 
  

   is 
  fiO/h, 
  and 
  so 
  

  

  velocity 
  of 
  H 
  = 
  — 
  £ 
  (1) 
  

  

  Thus, 
  to 
  the 
  factor 
  2ah/fi, 
  the 
  velocity 
  of 
  H 
  represents 
  the 
  

   magnitude 
  of 
  the 
  acceleration 
  of 
  the 
  planet. 
  Its 
  direction 
  is 
  

   at 
  right 
  angles 
  to 
  SP, 
  and 
  represents 
  the 
  direction 
  that 
  PS 
  

   would 
  have 
  if 
  SP 
  were 
  turned 
  90° 
  about 
  P 
  in 
  the 
  opposite 
  

   direction 
  to 
  that 
  in 
  which 
  SP 
  turns, 
  as 
  P 
  moves 
  in 
  the 
  

   orbit. 
  

  

  Drawing 
  the 
  chord 
  H 
  H, 
  which 
  on 
  the 
  scale 
  adopted 
  

   represents 
  the 
  total 
  change 
  of 
  velocity, 
  between 
  a 
  previous 
  

   point 
  H 
  , 
  corresponding 
  to 
  a 
  radius 
  vector 
  SP 
  , 
  and 
  H, 
  we 
  

   see 
  that, 
  if 
  we 
  choose 
  the 
  proper 
  point 
  in 
  the 
  plane 
  of 
  the 
  

   circle, 
  OH 
  and 
  OH 
  will 
  represent 
  the 
  velocities 
  at 
  the 
  

   beginning 
  and 
  end 
  of 
  the 
  time 
  t. 
  It 
  is 
  clear 
  that 
  must 
  lie 
  

   within 
  the 
  circle, 
  for 
  the 
  vector 
  OH, 
  which 
  represents 
  the 
  

   velocity, 
  must 
  turn 
  through 
  an 
  angle 
  2it 
  while 
  the 
  planet 
  

   traverses 
  the 
  orbit 
  once, 
  which 
  would 
  not 
  be 
  the 
  case 
  if 
  

   were 
  outside 
  the 
  circle. 
  

  

  The 
  position 
  of 
  must 
  be 
  independent 
  of 
  the 
  value 
  of 
  t 
  y 
  

   otherwise 
  the 
  speed 
  for 
  the 
  radius 
  vector 
  SP 
  would 
  depend 
  

   on 
  the 
  choice 
  of 
  the 
  initial 
  radius 
  vector 
  SP 
  . 
  Thus 
  is 
  a 
  

   definite 
  point. 
  This 
  result, 
  taken 
  along 
  with 
  the 
  repre- 
  

   sentation 
  of 
  the 
  acceleration 
  by 
  the 
  motion 
  of 
  a 
  point 
  in 
  

   the 
  circle, 
  shows 
  that 
  the 
  hodograph 
  of 
  the 
  planet 
  is 
  a 
  circle. 
  

   The 
  velocities 
  are 
  represented 
  in 
  magnitude 
  and 
  direction 
  by 
  

   the 
  radii 
  vectores 
  drawn 
  to 
  the 
  circle 
  from 
  the 
  eccentric 
  

   point 
  0. 
  

  

  OH 
  is 
  perpendicular 
  to 
  the 
  direction 
  of 
  motion 
  at 
  P 
  , 
  and 
  

   OH 
  to 
  the 
  direction 
  of 
  motion 
  at 
  P. 
  The 
  members 
  of 
  the 
  

   family 
  of 
  lines 
  drawn 
  from 
  to 
  the 
  sequence 
  of 
  points 
  H 
  

   are 
  perpendiculars 
  to 
  the 
  corresponding 
  tangents 
  to 
  the 
  

   ] 
  ath. 
  

  

  4. 
  It 
  is 
  obvious 
  that 
  OH 
  may 
  be 
  resolved 
  into 
  the 
  two- 
  

   components 
  OC, 
  CH, 
  that 
  is 
  into 
  two 
  components 
  of 
  fixed 
  

   amounts, 
  i\, 
  r 
  2 
  , 
  at 
  right 
  angles 
  respectively 
  to 
  tjje 
  line 
  00 
  

  

  