﻿426 
  MR 
  W. 
  J. 
  M. 
  RANKINE 
  ON 
  THE 
  CENTRIFUGAL 
  THEORY 
  OF 
  ELASTICITY, 
  

  

  tion, 
  that 
  the 
  atomic 
  atmospheres 
  might 
  be 
  treated 
  in 
  calculation 
  as 
  if 
  spherical, 
  

   did 
  not 
  give 
  rise 
  to 
  any 
  error. 
  

  

  By 
  the 
  aid 
  of 
  certain 
  transformations 
  in 
  those 
  equations, 
  I 
  have 
  been 
  enabled, 
  

   in 
  investigating 
  the 
  principles 
  of 
  the 
  mutual 
  transformation 
  of 
  heat 
  and 
  expansive 
  

   power, 
  to 
  deduce 
  Joule's 
  law 
  of 
  the 
  equivalence 
  of 
  heat 
  and 
  mechanical 
  power 
  

   directly 
  from 
  them, 
  instead 
  of 
  taking 
  it 
  (as 
  I 
  did 
  in 
  my 
  previous 
  papers) 
  as 
  a 
  con- 
  

   sequence 
  of 
  the 
  principle 
  of 
  vis-viva. 
  Carnot's 
  law 
  is 
  also 
  deduced 
  directly 
  from 
  

   the 
  hypothesis, 
  as 
  in 
  one 
  of 
  the 
  previous 
  papers. 
  

  

  (2.) 
  Classification 
  of 
  Elastic 
  Pressures. 
  — 
  The 
  pressures 
  considered 
  in 
  the 
  present 
  

   paper 
  are 
  those 
  only 
  which 
  depend 
  on 
  the 
  volume 
  occupied 
  by 
  a 
  given 
  weight 
  of 
  

   the 
  substance 
  ; 
  not 
  those 
  which 
  resist 
  change 
  of 
  figure 
  in 
  solids 
  and 
  viscous 
  liquids. 
  

   Certain 
  mathematical 
  relations 
  exist 
  between 
  those 
  two 
  classes 
  of 
  pressures 
  ; 
  but 
  

   they 
  do 
  not 
  affect 
  the 
  present 
  investigation. 
  

  

  To 
  illustrate 
  this 
  symbolically, 
  let 
  V 
  represent 
  the 
  volume 
  occupied 
  by 
  unity 
  

  

  of 
  weight 
  of 
  the 
  substance, 
  so 
  that 
  ^ 
  is 
  the 
  mean 
  density 
  ; 
  Q, 
  the 
  quantity 
  of 
  heat 
  

  

  in 
  unity 
  of 
  weight, 
  that 
  is 
  to 
  say, 
  the 
  vis- 
  viva 
  of 
  the 
  molecular 
  revolutions, 
  which, 
  

   according 
  to 
  the 
  hypothesis, 
  give 
  rise 
  to 
  the 
  expansive 
  pressure 
  depending 
  on 
  heat 
  ; 
  

   and 
  let 
  P 
  denote 
  the 
  total 
  expansive 
  pressure. 
  Then, 
  

  

  P 
  = 
  F(V,Q)+/(V) 
  • 
  ' 
  (10 
  

  

  In 
  this 
  equation, 
  F 
  (V, 
  Q) 
  is 
  the 
  pressure 
  of 
  the 
  atomic 
  atmospheres 
  at 
  the 
  sur- 
  

   faces 
  called 
  their 
  boundaries, 
  which 
  varies 
  with 
  the 
  centrifugal 
  force 
  of 
  the 
  mole- 
  

   cular 
  vortices 
  as 
  well 
  as 
  with 
  the 
  mean 
  density 
  ; 
  and/ 
  (V) 
  is 
  a 
  portion 
  of 
  pressure 
  

   due 
  to 
  the 
  mutual 
  attractions 
  and 
  repulsions 
  of 
  distinct 
  atoms, 
  and 
  varying 
  with 
  

   the 
  number 
  of 
  atoms 
  in 
  a 
  given 
  volume 
  only. 
  If 
  the 
  above 
  equation 
  be 
  differentiated 
  

   with 
  respect 
  to 
  the 
  hyberbolic 
  logarithm 
  of 
  the 
  density, 
  we 
  obtain 
  the 
  coefficient 
  

   of 
  elasticity 
  of 
  volume 
  

  

  6=-^=-^ 
  F 
  ^«-A/W 
  • 
  • 
  • 
  (ia.) 
  

  

  V 
  V 
  V 
  

  

  where 
  & 
  denotes 
  the 
  cubic 
  compressibility. 
  

  

  The 
  latter 
  portion 
  of 
  this 
  coefficient, 
  ~^y 
  /(V), 
  consists 
  of 
  two 
  parts, 
  one 
  of 
  

  

  V" 
  

   which 
  is 
  capable 
  of 
  being 
  resolved 
  into 
  forces, 
  acting 
  along 
  the 
  lines 
  joining 
  the 
  

   atomic 
  centres, 
  and 
  gives 
  rise 
  to 
  rigidity, 
  or 
  elasticity 
  of 
  figure, 
  as 
  well 
  to 
  elas- 
  

   ticity 
  of 
  volume, 
  while 
  the 
  other, 
  which 
  is 
  not 
  capable 
  of 
  being 
  so 
  resolved, 
  gives 
  

   rise 
  to 
  elasticity 
  of 
  volume 
  only. 
  The 
  ratio 
  of 
  each 
  of 
  those 
  parts 
  to 
  their 
  sum 
  

   must 
  be 
  a 
  function 
  of 
  the 
  heat, 
  the 
  former 
  part 
  being 
  greater, 
  and 
  the 
  latter 
  less, 
  

   as 
  the 
  atomic 
  atmosphere 
  is 
  more 
  concentrated 
  round 
  the 
  nucleus 
  ; 
  that 
  is 
  to 
  say, 
  

  

  