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  PROFESSOR 
  WILLIAM 
  THOMSON 
  ON 
  THE 
  

  

  in 
  different 
  states 
  at 
  the 
  same 
  temperature, 
  as 
  water 
  and 
  saturated 
  steam, 
  or 
  

   ice 
  and 
  water. 
  

  

  47. 
  In 
  the 
  first 
  place, 
  it 
  may 
  be 
  remarked, 
  that, 
  by 
  the 
  definition 
  of 
  M 
  and 
  N 
  

   in 
  § 
  20, 
  N 
  must 
  be 
  what 
  is 
  commonly 
  called 
  the 
  " 
  specific 
  heat 
  at 
  constant 
  volume" 
  

   of 
  the 
  substance, 
  provided 
  the 
  quantity 
  of 
  the 
  medium 
  be 
  the 
  standard 
  quantity 
  

   adopted 
  for 
  specific 
  heats, 
  which, 
  in 
  all 
  that 
  follows, 
  I 
  shall 
  take 
  as 
  the 
  unit 
  of 
  

   weight. 
  Hence 
  the 
  fundamental 
  equation 
  of 
  the 
  dynamical 
  theory, 
  (2) 
  of 
  § 
  20, 
  ex- 
  

   presses 
  a 
  relation 
  between 
  this 
  specific 
  heat 
  and 
  the 
  quantities 
  for 
  the 
  particular 
  

   substance 
  denoted 
  by 
  M 
  and 
  p. 
  If 
  we 
  eliminate 
  M 
  from 
  this 
  equation, 
  by 
  means 
  

   of 
  equation 
  (3) 
  of 
  § 
  21, 
  derived 
  from 
  the 
  expression 
  of 
  the 
  second 
  fundamental 
  

   principle 
  of 
  the 
  theory 
  of 
  the 
  motive 
  power 
  of 
  heat, 
  we 
  find 
  

  

  r/N 
  _ 
  \fJL 
  dt) 
  1 
  dp 
  

  

  dv 
  dt 
  J 
  dt 
  ' 
  [ 
  h 
  

  

  which 
  expresses 
  a 
  relation 
  between 
  the 
  variation 
  in 
  the 
  specific 
  heat 
  at 
  constant 
  

   volume, 
  of 
  any 
  substance 
  produced 
  by 
  an 
  alteration 
  of 
  its 
  volume 
  at 
  a 
  constant 
  

   temperature, 
  and 
  the 
  variation 
  of 
  its 
  pressure 
  with 
  its 
  temperature 
  when 
  the 
  

   volume 
  is 
  constant 
  ; 
  involving 
  a 
  function, 
  /jl, 
  of 
  the 
  temperature, 
  which 
  is 
  the 
  

   same 
  for 
  all 
  substances. 
  

  

  48. 
  Again, 
  let 
  K 
  denote 
  the 
  specific 
  heat 
  of 
  the 
  substance 
  under 
  constant 
  

   pressure. 
  Then, 
  if 
  dv 
  and 
  dt 
  be 
  so 
  related 
  that 
  the 
  pressure 
  of 
  the 
  medium 
  

   when 
  its 
  volume 
  and 
  temperature 
  are 
  v 
  + 
  dv 
  and 
  t+dt, 
  respectively, 
  is 
  the 
  same 
  

   as 
  when 
  they 
  are 
  v 
  and 
  t, 
  that 
  is, 
  if 
  

  

  = 
  ^dv 
  + 
  d 
  /dt; 
  ■ 
  

   dv 
  d 
  t 
  

  

  we 
  have 
  Kdt 
  = 
  Mdv 
  + 
  Ndt. 
  

  

  Hence 
  we 
  find 
  

  

  dp 
  

  

  M=-^(K-N) 
  (15), 
  

  

  Tt 
  

  

  which 
  merely 
  shews 
  the 
  meaning, 
  in 
  terms 
  of 
  the 
  two 
  specific 
  heats, 
  of 
  what 
  I 
  

   have 
  denoted 
  by 
  M. 
  Using 
  in 
  this 
  for 
  M 
  its 
  value 
  given 
  by 
  (3) 
  of 
  § 
  21, 
  we 
  find 
  

  

  m 
  

  

  K-W=-«^j 
  .... 
  (16), 
  

  

  ' 
  dv 
  

  

  an 
  expression 
  for 
  the 
  difference 
  between 
  the 
  two 
  specific 
  heats, 
  derived 
  without 
  

   hypothesis, 
  from 
  the 
  second 
  fundamental 
  principle 
  of 
  the 
  theory 
  of 
  the 
  motive 
  

   power 
  of 
  heat. 
  

  

  49. 
  These 
  results 
  may 
  be 
  put 
  into 
  forms 
  more 
  convenient 
  for 
  use, 
  in 
  applica- 
  

   tions 
  to 
  liquid 
  and 
  solid 
  media, 
  by 
  introducing 
  the 
  notation 
  : 
  — 
  

  

  