﻿340 
  Lord 
  Rayleigh 
  on 
  the 
  

  

  so 
  that 
  when 
  os 
  = 
  l 
  

  

  J=f(-4>i 
  + 
  2^-3<k+...). 
  

  

  Accordingly 
  the 
  force 
  tending 
  to 
  drive 
  out 
  the 
  ring 
  at 
  

   x 
  = 
  l 
  is 
  at 
  time 
  t 
  

  

  iw.j(-4> 
  1 
  +2<j> 
  2 
  -&<p 
  3 
  +...y, 
  

  

  or 
  in 
  the 
  mean 
  taken 
  over 
  a 
  long 
  interval, 
  

  

  iW.Mean2^f-V 
  2 
  . 
  

  

  Comparing 
  with 
  (5), 
  we 
  see 
  that 
  the 
  mean 
  force 
  L 
  has 
  the 
  

   value 
  2£xmeanV; 
  or 
  since 
  mean 
  V 
  = 
  mean 
  T=^-E, 
  E 
  de- 
  

   noting 
  the 
  constant 
  total 
  energy, 
  

  

  L=E/Z 
  (6) 
  

  

  The 
  force 
  driving 
  out 
  the 
  ring 
  is 
  thus 
  numerically 
  equal 
  to 
  

   the 
  longitudinal 
  density 
  of 
  the 
  energy. 
  

  

  This 
  result 
  may 
  readily 
  be 
  extended 
  to 
  cases 
  where 
  the 
  

   vibrations 
  are 
  not 
  limited 
  to 
  one 
  plane 
  ; 
  and 
  indeed 
  the 
  case 
  

   in 
  which 
  the 
  plane 
  of 
  the 
  string 
  uniformly 
  revolves 
  is 
  espe- 
  

   cially 
  simple 
  in 
  that 
  T 
  and 
  Y 
  are 
  then 
  constant 
  with 
  respect 
  

   to 
  time. 
  

  

  If 
  the 
  ring 
  is 
  allowed 
  to 
  move 
  out 
  slowly, 
  we 
  have 
  

  

  dE 
  = 
  -L^=-EiZ/7, 
  

   or 
  on 
  integration 
  

  

  E 
  = 
  E 
  1 
  Z- 
  1 
  , 
  (7) 
  

  

  analogous 
  to 
  (5), 
  though 
  different 
  from 
  it 
  in 
  the 
  power 
  of 
  I 
  

   involved. 
  If 
  I 
  increase 
  without 
  limit, 
  the 
  whole 
  energy 
  of 
  

   the 
  vibrations 
  may 
  be 
  abstracted 
  in 
  the 
  form 
  of 
  work 
  done 
  on 
  

   the 
  ring. 
  

  

  We 
  will 
  now 
  pass 
  on 
  to 
  consider 
  the 
  case 
  of 
  air 
  in 
  a 
  

   cylinder, 
  vibrating 
  in 
  one 
  dimension 
  and 
  supposed 
  to 
  obey 
  

   Boyle's 
  law 
  according 
  to 
  which 
  p 
  = 
  a 
  2 
  p. 
  By 
  the 
  general 
  

   hydrodynamical 
  equation 
  ( 
  4 
  Theory 
  of 
  Sound,' 
  § 
  253 
  a), 
  

  

  "-J 
  P 
  ~ 
  ~dT 
  * 
  u 
  ' 
  ••->•■(*) 
  

  

  where 
  </> 
  denotes 
  the 
  velocity-potential 
  and 
  U 
  the 
  resultant 
  

   velocity 
  at 
  any 
  point; 
  so 
  that 
  in 
  the 
  present 
  case, 
  if 
  we 
  

   integrate 
  over 
  a 
  long 
  interval 
  of 
  time, 
  

  

  a 
  2 
  Jlog^^ 
  + 
  ifU 
  2 
  ^ 
  ..... 
  (9) 
  

  

  