﻿Pressure-integral 
  as 
  Kinetic 
  Potential. 
  443 
  

  

  or 
  we 
  have 
  (4) 
  repeated 
  without 
  modification. 
  If 
  we 
  write 
  

   (14) 
  as 
  

  

  and 
  add 
  (16) 
  we 
  get 
  

  

  jV 
  * 
  + 
  J 
  j!>* 
  *■= 
  Jg?2» 
  2 
  - 
  l> 
  rfr- 
  (p 
  (I* 
  - 
  vjy 
  d«, 
  en) 
  

  

  a 
  result 
  differing 
  from 
  (5) 
  only 
  in 
  the 
  term 
  containing 
  

   intrinsic 
  energy. 
  

  

  A 
  change 
  in 
  the 
  position 
  of 
  a 
  barrier 
  a 
  gives 
  rise 
  to 
  

   no 
  ambiguity 
  in 
  the 
  kinetic 
  potential, 
  for 
  it 
  means 
  an 
  equal 
  

   change 
  in 
  the 
  estimate 
  of 
  each 
  side 
  of 
  equation 
  (17), 
  and 
  the 
  

  

  if 
  i 
  c* 
  

  

  added 
  term 
  has 
  the 
  form 
  -j 
  . 
  It 
  is 
  in 
  fact 
  -=- 
  I 
  pre 
  dr, 
  where 
  

  

  the 
  integral 
  extends 
  to 
  the 
  space 
  bounded 
  by 
  the 
  two 
  positions 
  

   of 
  the 
  barrier 
  and 
  a 
  strip 
  of 
  S. 
  

  

  For 
  the 
  case 
  of 
  sound 
  originated 
  by 
  the 
  vibration 
  of 
  a 
  

   solid 
  some 
  simplification 
  is 
  possible. 
  We 
  omit 
  k 
  in 
  (17) 
  

   write 
  p 
  = 
  p 
  (l-\-s) 
  and 
  treat 
  s 
  as 
  small. 
  We 
  have 
  then 
  

  

  while 
  -l* 
  = 
  m 
  s=T 
  2 
  

  

  at 
  po 
  

  

  and 
  the 
  second 
  term 
  in 
  pJJ 
  becomes 
  ~^ 
  2 
  [■>—) 
  . 
  This 
  taken 
  

   with 
  I 
  ^ 
  2u 
  2 
  dr, 
  if 
  p 
  is 
  put 
  for 
  p, 
  gives 
  

  

  2 
  J 
  IW 
  + 
  Byi 
  + 
  ^-l 
  -y^t\j 
  ar 
  - 
  

  

  In 
  applying 
  (13) 
  to 
  this 
  expression 
  we 
  may 
  omit 
  the 
  second 
  

   surface 
  integral 
  as 
  of 
  the 
  third 
  order 
  of 
  small 
  quantities, 
  and 
  

  

  J7\rh 
  

   (/> 
  ^- 
  dS. 
  The 
  first 
  part 
  

  

  of 
  pTJ 
  h\p 
  sdr, 
  or 
  as 
  ( 
  1 
  + 
  s) 
  dr 
  = 
  dr 
  element 
  of 
  original 
  

   volume, 
  if 
  p 
  Q 
  and 
  p 
  are 
  constant 
  the 
  integral 
  is 
  j9 
  \ 
  (dT 
  — 
  dr), 
  

  

  and 
  the 
  second 
  factor 
  is 
  the 
  total 
  diminution 
  in 
  volume 
  of 
  

   air. 
  Thus 
  if 
  we 
  alter 
  the 
  meaning 
  of 
  t 
  and 
  t 
  , 
  so 
  that 
  they 
  

   represent 
  the 
  whole 
  volume 
  of 
  the 
  solid, 
  r 
  corresponding 
  to 
  

  

  