﻿186 
  Prof. 
  A. 
  Gray 
  on 
  Hodographic 
  Treatment 
  and 
  

  

  7. 
  We 
  deduce 
  the 
  energy 
  equation 
  as 
  follows. 
  The 
  

   resultant 
  speed 
  v 
  is 
  given 
  in 
  terms 
  of 
  v 
  l9 
  v 
  2 
  by 
  

  

  v 
  2 
  = 
  v 
  1 
  2 
  +v 
  2 
  2 
  + 
  2v 
  1 
  V2COS0, 
  .... 
  (10) 
  

  

  which 
  by 
  (6) 
  and 
  (9) 
  can 
  be 
  written 
  

  

  2>.l- 
  

  

  e-t) 
  cu) 
  

  

  The 
  term 
  fjujr 
  represents 
  the 
  potential 
  energy 
  exhausted 
  by 
  

   the 
  passage 
  or" 
  the 
  planet 
  under 
  the 
  central 
  attraction 
  from 
  

   infinity 
  to 
  the 
  distance 
  r. 
  Writing 
  the 
  equation 
  in 
  the 
  

   form 
  

  

  £*-<!=-#, 
  (12) 
  

  

  2 
  r 
  la 
  v 
  y 
  

  

  and 
  taking 
  —fi/r 
  as 
  the 
  potential 
  energy, 
  we 
  see 
  that 
  the 
  

   kinetic 
  and 
  potential 
  energies 
  have 
  a 
  constant 
  negative 
  sum, 
  

   — 
  \x\2a. 
  From 
  this 
  equation 
  we 
  shall 
  draw 
  some 
  conclusions 
  

   which 
  appear 
  interesting. 
  [It 
  is 
  to 
  be 
  understood 
  that 
  when 
  

   £>1, 
  that 
  is 
  when 
  the 
  orbit 
  is 
  a 
  hyperbola, 
  the 
  sum 
  of 
  the 
  

   energies 
  ^v 
  2 
  and 
  —fi/r 
  is 
  + 
  fi/2a.] 
  

  

  The 
  constant 
  angular 
  momentum 
  is 
  fi/v 
  2 
  = 
  efi/vi=zb(/j,/a)*. 
  

   Hence 
  the 
  period 
  of 
  revolution 
  is 
  

  

  "*-*£/. 
  • 
  • 
  • 
  • 
  d3) 
  

  

  T 
  = 
  2: 
  

  

  and 
  is 
  therefore 
  independent 
  of 
  the 
  eccentricity. 
  

  

  8. 
  This 
  gives 
  an 
  interesting 
  instantaneous 
  solution 
  of 
  the 
  

   elementary 
  problem 
  of 
  polar 
  dynamics. 
  The 
  orbital 
  motion 
  

   of 
  a 
  planet 
  is 
  annulled 
  when 
  the 
  distance 
  from 
  the 
  sun 
  is 
  d 
  ; 
  

   find 
  the 
  time 
  which 
  the 
  planet 
  will 
  take 
  to 
  fall 
  to 
  the 
  centre 
  of 
  

   force. 
  Since 
  the 
  period 
  is 
  independent 
  of 
  the 
  eccentricity, 
  

   let 
  e 
  be 
  less 
  than 
  but 
  very 
  nearly 
  equal 
  to 
  1. 
  The 
  foci 
  are 
  

   practically 
  at 
  the 
  ends 
  of 
  the 
  major 
  axis. 
  When 
  the 
  planet 
  

   has 
  just 
  passed 
  (not 
  rounded) 
  the 
  aphelion 
  end, 
  the 
  speed 
  

   along 
  the 
  major 
  axis 
  is 
  zero. 
  The 
  time 
  taken 
  to 
  reach 
  the 
  

   other 
  focus 
  is 
  half 
  the 
  period, 
  and 
  so 
  the 
  time 
  taken 
  by 
  

   the 
  planet 
  to 
  fall 
  into 
  the 
  sun 
  is 
  ir(ld 
  s 
  /fjb)i, 
  as 
  may 
  be 
  

   verified 
  at 
  once 
  by 
  direct 
  integration. 
  

  

  The 
  earth 
  in 
  the 
  circumstances 
  stated 
  would 
  fall 
  into 
  the 
  

   sun 
  in 
  -|T/2\/2, 
  that 
  is 
  in 
  about 
  65 
  days. 
  

  

  [It 
  may 
  be 
  noticed 
  incidentally 
  that 
  if 
  we 
  use 
  the 
  value 
  of 
  

   v 
  2 
  given 
  on 
  the 
  right 
  of 
  (11), 
  with 
  the 
  values 
  of 
  v 
  l5 
  v 
  2 
  given 
  

   in 
  (9) 
  and 
  (10), 
  and 
  the 
  values 
  of 
  the 
  action 
  and 
  the 
  period 
  

   just 
  found, 
  we 
  obtain 
  after 
  a 
  little 
  reduction 
  the 
  integral 
  

   >2 
  l_cose_d0__ 
  e 
  

  

  , 
  o 
  (l 
  + 
  .cos0) 
  2 
  - 
  ^(l-^fJ 
  

  

  I 
  

  

  