﻿184 
  Prof. 
  A. 
  Gray 
  on 
  Hodographic 
  Treatment 
  and 
  

  

  and 
  to 
  the 
  radius 
  vector 
  SP. 
  We 
  can 
  now 
  instantaneously 
  

   find 
  the 
  orbit. 
  

  

  Let 
  be 
  the 
  angle 
  which 
  SP 
  makes 
  with 
  the 
  direction 
  OC. 
  

   The 
  angular 
  momentum 
  of 
  the 
  planet 
  about 
  S 
  is 
  r( 
  v 
  2 
  + 
  Vxcostf), 
  

   which 
  has 
  a 
  constant 
  value 
  h. 
  Thus 
  if 
  we 
  write 
  e 
  for 
  v 
  1 
  /v 
  2 
  , 
  

   we 
  get 
  

  

  that 
  is 
  the 
  path 
  is 
  a 
  conic 
  section 
  of 
  which 
  S 
  is 
  a 
  focus. 
  

  

  The 
  length 
  of 
  the 
  major 
  axis 
  is 
  the 
  sum 
  of 
  the 
  lengths 
  ri, 
  r 
  2 
  

   of 
  r 
  obtained 
  by 
  putting 
  cos 
  = 
  1, 
  cos#= 
  — 
  1, 
  respectively. 
  

   If 
  it 
  is 
  denoted 
  by 
  2a 
  we 
  have 
  

  

  fcjfl+l) 
  ». 
  ... 
  (3) 
  

  

  Thus 
  the 
  perihelion 
  and 
  aphelion 
  distances 
  are 
  a(l 
  — 
  e) 
  and 
  

   a(l 
  + 
  e). 
  

  

  5. 
  It 
  is 
  convenient 
  to 
  make 
  the 
  radius 
  of 
  the 
  hodograph 
  

   equal 
  to 
  the 
  2a 
  just 
  found. 
  The 
  length 
  of 
  the 
  line 
  OH, 
  

   representing 
  the 
  velocity, 
  is 
  then 
  equal 
  to 
  twice 
  the 
  length 
  of 
  

   the 
  perpendicular 
  let 
  fall 
  from 
  on 
  a 
  line 
  which 
  is 
  parallel 
  

   to 
  the 
  tangent 
  to 
  the 
  orbit 
  at 
  the 
  corresponding 
  position 
  of 
  

   the 
  planet. 
  In 
  fact 
  a 
  circle 
  of 
  radius 
  2a 
  described 
  from 
  S 
  as 
  

   centre 
  serves 
  very 
  conveniently 
  as 
  hodograph. 
  The 
  hodo- 
  

   graphic 
  origin 
  is 
  then 
  coincident 
  with 
  the 
  empty 
  focus, 
  as 
  

   will 
  easily 
  be 
  seen 
  from 
  the 
  fact 
  that 
  the 
  perpendicular 
  from 
  

   the 
  empty 
  focus 
  to 
  any 
  tangent 
  of 
  the 
  ellipse 
  at 
  a 
  point 
  P 
  

   intersects 
  the 
  radius 
  vector 
  SP 
  on 
  the 
  circle. 
  

  

  6. 
  As 
  has 
  been 
  seen 
  in 
  (2) 
  the 
  two 
  components 
  v 
  1? 
  v 
  2 
  of 
  

   velocity 
  give 
  an 
  instantaneous 
  integration 
  of 
  the 
  differential 
  

   equations 
  of 
  the 
  orbit. 
  The 
  radial 
  differential 
  equation 
  

  

  r 
  -**=-£ 
  (4) 
  

  

  shows 
  this 
  more 
  clearly, 
  and 
  incidentally 
  gives 
  the 
  value 
  of 
  h 
  

   in 
  terms 
  of 
  v 
  2 
  . 
  Since 
  the 
  component 
  v 
  x 
  is 
  at 
  right 
  angles 
  to 
  

   the 
  major 
  axis, 
  and 
  is 
  directed 
  towards 
  the 
  side 
  of 
  that 
  axis 
  

   to'jwhich 
  the 
  planet 
  passes 
  at 
  perihelion, 
  we 
  have 
  

  

  r 
  = 
  ^i 
  sin 
  6, 
  = 
  — 
  , 
  .... 
  (5) 
  

  

  where 
  6 
  is 
  the 
  angle 
  traversed 
  by 
  the 
  radius 
  vector 
  from 
  

   perihelion 
  to 
  the 
  position 
  considered. 
  Hence 
  

  

  r 
  = 
  Vi 
  cos 
  6 
  . 
  6 
  = 
  v 
  x 
  cos 
  0(v 
  2 
  + 
  v 
  1 
  cos 
  6)/r, 
  

  

  