﻿342 
  Lord 
  Rayleigh 
  on 
  the 
  

  

  due 
  to 
  the 
  vibrations 
  and 
  tending 
  to 
  push 
  the 
  piston 
  out, 
  

  

  L=EZ- 
  1 
  . 
  (14) 
  

  

  As 
  in 
  the 
  case 
  of 
  the 
  string, 
  the 
  total 
  force 
  is 
  measured 
  by 
  

   the 
  longitudinal 
  density 
  of 
  the 
  total 
  energy; 
  or, 
  if 
  we 
  prefer 
  

   so 
  to 
  express 
  it, 
  the 
  additional 
  pressure 
  islneasured 
  by 
  the 
  

   volume-density 
  of 
  the 
  energy. 
  

  

  In 
  the 
  last 
  problem, 
  as 
  well 
  as 
  in 
  that 
  of 
  the 
  string, 
  the 
  

   vibrations 
  are 
  in 
  one 
  dimension. 
  In 
  the 
  case 
  of 
  air 
  there 
  is 
  

   no 
  difficulty 
  in 
  the 
  extension 
  to 
  two 
  or 
  three 
  dimensions. 
  

   Thus, 
  if 
  aerial 
  vibrations 
  be 
  distributed 
  equally 
  in 
  all 
  di- 
  

   rections, 
  the 
  pressure 
  due 
  to 
  them 
  coincides 
  with 
  one-third 
  of 
  

   the 
  volume-density 
  of 
  the 
  energy. 
  In 
  the 
  case 
  of 
  the 
  string, 
  

   where 
  the 
  vibrations 
  are 
  transverse, 
  we 
  cannot 
  find 
  an 
  

   analogue 
  in 
  three 
  dimensions; 
  but 
  a 
  membrane 
  with 
  a 
  flexible 
  

   and 
  extensible 
  boundary 
  capable 
  of 
  slipping 
  along 
  the 
  sur- 
  

   face, 
  provides 
  for 
  two 
  dimensions. 
  If 
  the 
  vibrations 
  be 
  

   equally 
  distributed 
  in 
  the 
  plane, 
  the 
  force 
  outwards 
  per 
  unit 
  

   length 
  of 
  contour 
  will 
  be 
  measured 
  by 
  one-half 
  of 
  the 
  super- 
  

   ficial 
  density 
  of 
  the 
  total 
  energy. 
  

  

  A 
  more 
  general 
  treatment 
  of 
  the 
  question 
  may 
  be 
  effected 
  

   by 
  means 
  of 
  Lagrange's 
  theory. 
  If 
  I 
  be 
  one 
  of 
  the 
  coordi- 
  

   nates 
  fixing 
  the 
  configuration 
  of 
  a 
  system, 
  the 
  corresponding 
  

   equation 
  is 
  

  

  d/dl\ 
  OF 
  dV 
  .... 
  

  

  dt{dF)-M 
  + 
  -di= 
  L 
  > 
  ■ 
  ■ 
  ■ 
  • 
  W 
  

  

  where 
  T 
  and 
  Y 
  denote 
  as 
  usual 
  the 
  expressions 
  for 
  the 
  kinetic 
  

   and 
  potential 
  energies. 
  On 
  integration 
  over 
  a 
  time 
  ^ 
  

  

  fLtfe.l 
  rdT-i 
  If 
  A*V_dT\ 
  , 
  

   J 
  h 
  tXdV 
  \ 
  h)\dl 
  dl 
  )■ 
  

  

  If 
  dT/dl' 
  remain 
  finite 
  throughout, 
  and 
  if 
  the 
  range 
  of 
  integra- 
  

   tion 
  be 
  sufficiently 
  extended, 
  the 
  integrated 
  term 
  disappears, 
  

   and 
  we 
  get 
  

  

  [Ldt 
  11/dY 
  dT\ 
  Jt 
  , 
  1C 
  , 
  

  

  On 
  the 
  right 
  hand 
  of 
  (16) 
  the 
  differentiations 
  are 
  partial, 
  

   the 
  coordinates 
  other 
  than 
  / 
  and 
  all 
  the 
  velocities 
  being 
  sup- 
  

   posed 
  constant. 
  

  

  We 
  will 
  apply 
  our 
  equation 
  (16) 
  in 
  the 
  first 
  place 
  to 
  the 
  

   simple 
  pendulum 
  of 
  fig. 
  1, 
  I 
  denoting 
  the 
  length 
  of 
  the 
  

   vibrating 
  portion 
  of 
  the 
  string 
  BC. 
  If 
  a, 
  y 
  be 
  the 
  horizontal 
  

  

  