﻿DYNAMICAL 
  THEORY 
  OF 
  HEAT. 
  481 
  

  

  data 
  from 
  which 
  it 
  is 
  deduced. 
  It 
  is 
  not 
  improbable 
  that 
  such 
  a 
  Table 
  or 
  Empi- 
  

   rical 
  Function, 
  and 
  a 
  similar 
  representation 
  of 
  the 
  pressure, 
  may 
  be 
  found 
  to 
  be 
  

   the 
  most 
  convenient 
  expression 
  for 
  results 
  of 
  complete 
  observations 
  on 
  the 
  com- 
  • 
  

   possibility, 
  the 
  law 
  of 
  expansion 
  by 
  heat, 
  and 
  the 
  thermal 
  capacities 
  of 
  a 
  vapour 
  

   or 
  gas. 
  

  

  94. 
  The 
  principles 
  brought 
  forward 
  in 
  a 
  former 
  communication 
  " 
  On 
  a 
  Means 
  

   of 
  discovering 
  experimentally, 
  &c." 
  (which 
  is 
  now 
  referred 
  to 
  as 
  Part 
  IV. 
  of 
  a 
  series 
  

   of 
  papers 
  on 
  the 
  Dynamical 
  Theory 
  of 
  Heat), 
  may 
  be 
  expressed 
  in 
  a 
  more 
  con- 
  

   venient, 
  and 
  in 
  a 
  somewhat 
  more 
  comprehensive 
  manner 
  than 
  in 
  the 
  formulae 
  

   contained 
  in 
  that 
  paper, 
  by 
  introducing 
  the 
  notations 
  and 
  principles 
  which 
  form 
  

   the 
  subject 
  of 
  the 
  present 
  communication. 
  Thus, 
  let 
  t 
  be 
  the 
  temperature, 
  and 
  a 
  

   the 
  volume 
  of 
  a 
  pound, 
  of 
  air 
  flowing 
  gently 
  in 
  a 
  pipe 
  (under 
  very 
  high 
  pressure 
  it 
  

   maybe) 
  towards 
  a 
  very 
  narrow 
  passage 
  (a 
  nearly 
  closed 
  stopcock, 
  for 
  instance), 
  and 
  

   let 
  p 
  be 
  its 
  pressure. 
  Let 
  f, 
  u', 
  and 
  p' 
  be 
  the 
  corresponding 
  qualities 
  of 
  the 
  air, 
  

   flowing 
  gently 
  through 
  a 
  continuation 
  of 
  the 
  pipe, 
  after 
  having 
  passed 
  the 
  " 
  rapids'" 
  

   in 
  and 
  near 
  the 
  narrow 
  passage. 
  Let 
  Q 
  be 
  the 
  quantity 
  of 
  heat 
  (which, 
  according 
  

   to 
  circumstances, 
  may 
  be 
  positive, 
  zero, 
  or 
  negative) 
  emitted 
  by 
  a 
  pound 
  of 
  air 
  

   during 
  its 
  whole 
  passage 
  from 
  the 
  former 
  locality 
  through 
  the 
  narrow 
  passage, 
  to 
  

   the 
  latter 
  ; 
  and 
  let 
  S 
  denote 
  the 
  mechanical 
  value 
  of 
  the 
  sound 
  emitted 
  from 
  the 
  

   " 
  rapids." 
  The 
  only 
  other 
  external 
  mechanical 
  effect, 
  besides 
  these 
  two, 
  produced 
  

   by 
  the 
  air, 
  is 
  the 
  excess 
  (which, 
  according 
  to 
  circumstances, 
  may 
  be 
  negative, 
  zero, 
  

   or 
  positive) 
  of 
  the 
  work 
  done 
  by 
  the 
  air 
  in 
  pressing 
  out 
  through 
  the 
  second 
  part 
  

   of 
  the 
  pipe 
  above 
  that 
  spent 
  in 
  pressing 
  it 
  in 
  through 
  the 
  first 
  ; 
  the 
  amount 
  of 
  

   which, 
  for 
  each 
  pound 
  of 
  air 
  that 
  passes, 
  is 
  of 
  course 
  p' 
  u'-p 
  u. 
  Hence, 
  the 
  

   whole 
  mechanical 
  value 
  of 
  the 
  effects 
  produced 
  externally 
  by 
  each 
  pound 
  of 
  the 
  

   ah-, 
  from 
  its 
  own 
  mechanical 
  energy, 
  is 
  

  

  J 
  Q+S+jt/w'— 
  pu, 
  (15). 
  

  

  Hence, 
  if 
  (v, 
  t) 
  denote 
  the 
  value 
  of 
  e 
  expressed 
  as 
  a 
  function 
  of 
  the 
  independent 
  

   variables, 
  v 
  and 
  t 
  ; 
  so 
  that 
  cp 
  {u, 
  t) 
  may 
  express 
  the 
  mechanical 
  energy 
  of 
  a 
  pound 
  

   of 
  ah 
  before, 
  and 
  </> 
  {u', 
  tf) 
  the 
  mechanical 
  energy 
  of 
  a 
  pound 
  of 
  air 
  after, 
  passing 
  

   the 
  rapids 
  ; 
  we 
  have 
  

  

  (/> 
  (< 
  f) 
  = 
  cp 
  (v,t)-{J 
  Q+S+p'u'-pu] 
  .... 
  (16). 
  

  

  95. 
  If 
  the 
  circumstances 
  be 
  arranged 
  (as 
  is 
  always 
  possible), 
  so 
  as 
  to 
  prevent 
  

   the 
  air 
  from 
  experiencing 
  either 
  gain 
  or 
  loss 
  of 
  heat 
  by 
  conduction 
  through 
  the 
  

   pipe 
  and 
  stopcock, 
  we 
  shall 
  have 
  Q=0; 
  and 
  if 
  (as 
  is 
  perhaps 
  also 
  possible) 
  only 
  

   a 
  mechanically 
  inappreciable 
  amount 
  of 
  sound 
  be 
  allowed 
  to 
  escape, 
  we 
  may 
  take 
  

   S=0. 
  Then 
  the 
  preceding 
  equation 
  becomes' 
  

  

  (f>(u',f) 
  = 
  (t)(u,t)-(p'u'-pu) 
  (17). 
  

  

  