﻿482 
  PROFESSOR 
  W. 
  THOMSON 
  ON 
  THE 
  DYNAMICAL 
  THEORY 
  OF 
  HEAT. 
  

  

  If 
  by 
  experimenting 
  in 
  such 
  circumstances 
  it 
  be 
  found 
  that 
  f 
  does 
  not 
  differ 
  sen- 
  

   sibly 
  from 
  t, 
  Mayer's 
  hypothesis 
  is 
  verified 
  for 
  air 
  at 
  the 
  temperature 
  t, 
  and, 
  as 
  

   •// 
  u' 
  would 
  then 
  be 
  equal 
  to 
  p 
  u, 
  according 
  to 
  Boyle 
  and 
  Mariotte's 
  law, 
  we 
  

   should 
  have 
  

  

  (p 
  (id, 
  t) 
  = 
  (j) 
  (u, 
  t) 
  

  

  which 
  is 
  in 
  fact 
  the 
  expression 
  of 
  Mayer's 
  hypothesis, 
  in 
  terms 
  of 
  the 
  notation 
  

   for 
  mechanical 
  energy 
  introduced 
  in 
  this 
  paper. 
  If, 
  on 
  the 
  other 
  hand, 
  f 
  be 
  found 
  

   to 
  differ 
  from 
  t 
  ;* 
  let 
  values 
  of 
  p, 
  pf, 
  t, 
  and 
  if 
  be 
  observed 
  in 
  various 
  experiments 
  

   of 
  this 
  kind, 
  and, 
  from 
  the 
  known 
  laws 
  of 
  density 
  of 
  air, 
  let 
  u 
  and 
  u' 
  be 
  calculated. 
  

   We 
  then 
  have, 
  by 
  an 
  application 
  of 
  (13), 
  to 
  the 
  results 
  of 
  each 
  experiment, 
  an 
  

   equation 
  shewing 
  the 
  difference 
  between 
  the 
  mechanical 
  energies 
  of 
  a 
  pound 
  of 
  

   air 
  in 
  two 
  particular 
  specified 
  states 
  as 
  to 
  temperature 
  and 
  density. 
  All 
  the 
  par- 
  

   ticular 
  equations 
  thus 
  obtained, 
  may 
  be 
  used 
  towards 
  forming, 
  or 
  for 
  correcting, 
  

   a 
  table 
  of 
  the 
  values 
  of 
  the 
  mechanical 
  energy 
  of 
  a 
  mass 
  of 
  air, 
  at 
  various 
  tempera- 
  

   tures 
  and 
  densities. 
  

  

  .96. 
  If, 
  according 
  to 
  the 
  plan 
  proposed 
  in 
  my 
  former 
  communication 
  (§ 
  72), 
  the 
  

   air, 
  on 
  leaving 
  the 
  narrow 
  passage, 
  be 
  made 
  to 
  pass 
  through 
  a 
  spiral 
  pipe 
  immersed 
  

   in 
  water 
  in 
  a 
  calorimetrical 
  apparatus, 
  and 
  be 
  so 
  brought 
  back 
  exactly 
  to 
  the 
  pri- 
  

   mitive 
  temperature 
  t, 
  we 
  should 
  have, 
  according 
  to 
  Boyle's 
  and 
  Mariotte's 
  law, 
  

   p' 
  u'-pu=0; 
  and 
  if 
  H 
  denote 
  the 
  value 
  of 
  Q, 
  in 
  this 
  particular 
  case 
  (or 
  the 
  

   quantity 
  of 
  heat 
  measured 
  by 
  means 
  of 
  the 
  calorimetric 
  apparatus), 
  the 
  general 
  

   equation 
  (16) 
  takes 
  the 
  form, 
  

  

  <p 
  (u\ 
  t) 
  = 
  (p 
  («,/)-(JH 
  + 
  S) 
  (18). 
  

  

  If 
  in 
  this 
  we 
  neglect 
  S, 
  as 
  probably 
  insensible, 
  and 
  if 
  we 
  substitute 
  for 
  <p 
  (u, 
  t) 
  

   and 
  <£ 
  {u 
  f 
  , 
  t) 
  expressions 
  deduced 
  from 
  (9), 
  we 
  find, 
  

  

  H=f* 
  7t-"^e^) 
  pulog— 
  (19), 
  

  

  which 
  agrees 
  exactly 
  with 
  the 
  expression 
  obtained 
  by 
  a 
  synthetical 
  process, 
  

   founded 
  on 
  the 
  same 
  principles, 
  in 
  my 
  former 
  communication 
  (§ 
  76). 
  

  

  * 
  If 
  the 
  values 
  of 
  ^ 
  I 
  have 
  used 
  formerly 
  be 
  correct, 
  t' 
  would 
  be 
  less 
  than 
  t, 
  for 
  all 
  cases 
  in 
  

   which 
  t 
  is 
  lower 
  than 
  about 
  30° 
  cent. 
  ; 
  but 
  on 
  the 
  contrary, 
  if 
  t 
  be 
  considerably 
  above 
  30° 
  cent., 
  t' 
  

   would 
  be 
  found 
  to 
  exceed 
  t. 
  (See 
  Account 
  of 
  Carnot's 
  Theory, 
  Appendix 
  II.) 
  It 
  may 
  be 
  shewn, 
  

   that 
  if 
  they 
  are 
  correct, 
  air 
  at 
  the 
  temperature 
  0° 
  forced 
  up 
  with 
  a 
  pressure 
  of 
  ten 
  atmospheres 
  towards 
  

   a 
  small 
  orifice, 
  and 
  expanding 
  through 
  it 
  to 
  the 
  atmospheric 
  pressure, 
  would 
  go 
  down 
  in 
  temperature 
  

   by 
  about 
  4 
  0- 
  4 
  ; 
  but 
  that 
  if 
  it 
  had 
  the 
  temperature 
  of 
  100° 
  in 
  approaching 
  the 
  orifice, 
  it 
  would 
  leave 
  

   at 
  a 
  temperature 
  about 
  5°-2 
  higher 
  ; 
  provided 
  that 
  in 
  each 
  case 
  there 
  is 
  no 
  appreciable 
  expenditure 
  

   of 
  mechanical 
  energy 
  on 
  sound. 
  

  

  