﻿DYNAMICAL 
  THEORY 
  OF 
  HEAT. 
  283 
  

  

  dp 
  

  

  (17), 
  

  

  K 
  = 
  V 
  X 
  — 
  . 
  

  

  CIV 
  

  

  1 
  dp 
  

  

  e 
  = 
  - 
  -r- 
  

  

  K 
  dt 
  

  

  where 
  k 
  will 
  be 
  the 
  reciprocal 
  of 
  the 
  compressibility, 
  and 
  e 
  the 
  coefficient 
  of 
  ex- 
  

   pansion 
  with 
  heat. 
  

  

  Equations 
  (14), 
  (16), 
  and 
  (3), 
  thus 
  become 
  

  

  GO 
  

  

  dK 
  _ 
  \pj 
  Ke 
  

  

  dv 
  - 
  dt 
  J 
  < 
  18) 
  ' 
  

  

  K 
  e 
  2 
  

  

  K-N 
  = 
  ^ 
  (19), 
  

  

  r" 
  

  

  M 
  = 
  j 
  Ke 
  (20); 
  

  

  r 
  

  

  the 
  third 
  of 
  these 
  equations 
  being 
  annexed 
  to 
  shew 
  explicitly 
  the 
  quantity 
  of 
  

   heat 
  developed 
  by 
  the 
  compression 
  of 
  the 
  substance 
  kept 
  at 
  a 
  constant 
  tempera- 
  

   ture. 
  Lastly, 
  if 
  6 
  denote 
  the 
  rise 
  in 
  temperature 
  produced 
  by 
  a 
  compression 
  from 
  

   v 
  + 
  d 
  v 
  to 
  v, 
  before 
  any 
  heat 
  is 
  emitted, 
  we 
  have 
  

  

  6 
  = 
  ~ 
  .— 
  .dv= 
  " 
  e 
  — 
  2 
  dv 
  (21). 
  

  

  N 
  fJL 
  fxK—vKe 
  2 
  K 
  J 
  

  

  50. 
  The 
  first 
  of 
  these 
  expressions 
  for 
  shews 
  that, 
  when 
  the 
  substance 
  con- 
  

   tracts 
  as 
  its 
  temperature 
  rises 
  (as 
  is 
  the 
  case, 
  for 
  instance, 
  with 
  water 
  between 
  

   its 
  freezing 
  point 
  and 
  its 
  point 
  of 
  maximum 
  density), 
  its 
  temperature 
  would 
  

   become 
  lowered 
  by 
  a 
  sudden 
  compression. 
  The 
  second, 
  which 
  shews, 
  in 
  terms 
  of 
  

   its 
  compressibility 
  and 
  expansibility, 
  exactly 
  how 
  much 
  the 
  temperature 
  of 
  any 
  

   substance 
  is 
  altered 
  by 
  an 
  infinitely 
  small 
  alteration 
  of 
  its 
  volume, 
  leads 
  to 
  the 
  

   approximate 
  expression 
  

  

  A 
  Ke 
  

  

  if, 
  as 
  is 
  probably 
  the 
  case 
  for 
  all 
  known 
  solids 
  and 
  liquids, 
  e 
  be 
  so 
  small 
  that 
  

   e 
  . 
  vne 
  is 
  very 
  small 
  compared 
  with 
  fxli. 
  

  

  51. 
  If, 
  now, 
  we 
  suppose 
  the 
  substance 
  to 
  be 
  a 
  gas, 
  and 
  introduce 
  the 
  hypo- 
  

   thesis 
  that 
  its 
  density 
  is 
  strictly 
  subject 
  to 
  the 
  " 
  gaseous 
  laws," 
  we 
  should 
  have, 
  

   by 
  Boyle 
  and 
  Mariotte's 
  law 
  of 
  compression, 
  

  

  d 
  / 
  = 
  - 
  P 
  - 
  (22); 
  

  

  dv 
  v 
  K 
  ' 
  ' 
  

  

  and 
  by 
  Dalton 
  and 
  Gay 
  Lussac's 
  law 
  of 
  expansion, 
  

  

  dv 
  E 
  v 
  

  

  dt 
  " 
  1 
  + 
  E* 
  

   from 
  which 
  we 
  deduce 
  

  

  dp 
  Ej» 
  

  

  d~t 
  ' 
  lTEl 
  ' 
  

  

  (23;; 
  

  

  