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  PROFESSOR 
  WILLIAM 
  THOMSON 
  ON 
  THE 
  

  

  that 
  the 
  true 
  value 
  of 
  fx 
  might 
  be 
  " 
  inversely 
  at 
  the 
  temperatures 
  from 
  zero;" 
  * 
  

   and 
  values 
  for 
  various 
  temperatures 
  calculated 
  by 
  means 
  of 
  the 
  formula, 
  

  

  ^ 
  = 
  J 
  i^ 
  HI). 
  

  

  were 
  given 
  for 
  comparison 
  with 
  those 
  which 
  I 
  had 
  calculated 
  from 
  data 
  regarding 
  

   steam. 
  This 
  formula 
  is 
  also 
  adopted 
  by 
  Clausius, 
  who 
  uses 
  it 
  fundamentally 
  in 
  

   his 
  mathematical 
  investigations. 
  If 
  /jl 
  were 
  correctly 
  expressed 
  by 
  it, 
  we 
  should 
  

   have 
  

  

  7 
  r 
  'S 
  1 
  + 
  ES 
  

  

  T 
  ^'= 
  J1 
  °SfTET 
  ; 
  

  

  and 
  therefore 
  equations 
  (1) 
  and 
  (2) 
  would 
  become 
  

  

  W=J^ 
  (12), 
  

  

  E 
  + 
  S 
  

  

  - 
  X 
  + 
  T 
  

   R 
  = 
  ^ 
  (13). 
  

  

  43. 
  The 
  reasons 
  upon 
  which 
  Mr 
  Joule's 
  opinion 
  is 
  founded, 
  that 
  the 
  preceding 
  

   equation 
  (11) 
  may 
  be 
  the 
  correct 
  expression 
  for 
  Carnot's 
  function, 
  although 
  the 
  

   values 
  calculated 
  by 
  means 
  of 
  it 
  differ 
  considerably 
  from 
  those 
  shewn 
  in 
  Table 
  I. 
  

   of 
  my 
  former 
  paper, 
  form 
  the 
  subject 
  of 
  a 
  communication, 
  which 
  I 
  hope 
  to 
  have 
  

   an 
  opportunity 
  of 
  laying 
  before 
  the 
  Royal 
  Society 
  previously 
  to 
  the 
  close 
  of 
  

   the 
  present 
  session. 
  

  

  Part 
  III. 
  — 
  Applications 
  of 
  the 
  Dynamical 
  Theory 
  to 
  establish 
  Relations 
  

  

  BETWEEN 
  THE 
  PHYSICAL 
  PROPERTIES 
  OF 
  ALL 
  SUBSTANCES. 
  

  

  44. 
  The 
  two 
  fundamental 
  equations 
  of 
  the 
  dynamical 
  theory 
  of 
  heat, 
  inves- 
  

   tigated 
  above, 
  express 
  relations 
  between 
  quantities 
  of 
  heat 
  required 
  to 
  pro- 
  

   duce 
  changes 
  of 
  volume 
  and 
  temperature 
  in 
  any 
  material 
  medium 
  whatever, 
  

  

  E 
  

   * 
  If 
  we 
  take 
  fx 
  = 
  k 
  - 
  — 
  =- 
  where 
  k 
  may 
  be 
  any 
  constant, 
  we 
  find 
  

  

  -'(rrJ 
  

  

  which 
  is 
  the 
  formula 
  I 
  gave 
  when 
  this 
  paper 
  was 
  communicated. 
  I 
  have 
  since 
  remarked, 
  that 
  Mr 
  

   Joule's 
  hypothesis 
  implies 
  essentially, 
  that 
  the 
  coefficient 
  k 
  must 
  be 
  as 
  it 
  is 
  taken 
  in 
  the 
  text, 
  the 
  

   mechanical 
  equivalent 
  of 
  a 
  thermal 
  unit. 
  Mr 
  Rankine, 
  in 
  a 
  letter 
  dated 
  March 
  27, 
  1851, 
  informs 
  

   me 
  that 
  he 
  has 
  deduced, 
  from 
  the 
  principles 
  laid 
  down 
  in 
  his 
  paper 
  communicated 
  last 
  year 
  to 
  this 
  

   Society, 
  an 
  approximate 
  formula 
  for 
  the 
  ratio 
  of 
  the 
  maximum 
  quantity 
  of 
  heat 
  converted 
  into 
  

   mechanical 
  effect 
  to 
  the 
  whole 
  quantity 
  expended, 
  in 
  an 
  expansive 
  engine 
  of 
  any 
  substance, 
  which, 
  

   on 
  comparison, 
  I 
  find 
  agrees 
  exactly 
  with 
  the 
  expression 
  (12) 
  given 
  in 
  the 
  text 
  as 
  a 
  consequence 
  of 
  

   the 
  hypothesis 
  suggested 
  by 
  Mr 
  Joule 
  regarding 
  the 
  value 
  of 
  /a 
  at 
  any 
  temperature. 
  — 
  [April 
  4, 
  1851.] 
  

  

  