﻿160 
  MR 
  W. 
  J. 
  M. 
  RANKINE 
  ON 
  THE 
  

  

  ,. 
  . 
  , 
  8H 
  1 
  SD 
  , 
  4ttR 
  3 
  D 
  „ 
  ., 
  . 
  . 
  

  

  which, 
  because 
  -^-== 
  — 
  k 
  • 
  --ft, 
  and 
  — 
  „ 
  — 
  =M 
  is 
  equal 
  to 
  

  

  We 
  must 
  suppose 
  that 
  the 
  velocity 
  of 
  oscillation 
  is 
  equalised 
  throughout 
  the 
  

   atomic 
  atmosphere, 
  by 
  a 
  propagation 
  of 
  motion 
  so 
  rapid 
  as 
  to 
  be 
  practically 
  

   instantaneous. 
  

  

  Then 
  if 
  the 
  above 
  expression 
  be 
  integrated 
  with 
  respect 
  to 
  du, 
  from 
  u=0 
  to 
  

   u=l, 
  the 
  result 
  will 
  give 
  the 
  whole 
  increase 
  of 
  heat 
  in 
  the 
  atom 
  arising 
  from 
  the 
  con- 
  

   densation 
  8 
  D 
  ; 
  and 
  dividing 
  that 
  integral 
  by 
  the 
  atomic 
  weight 
  M, 
  we 
  shall 
  obtain 
  

   the 
  corresponding 
  development 
  of 
  heat 
  in 
  unity 
  of 
  weight. 
  This 
  is 
  expressed 
  by 
  

   the 
  following 
  equation 
  : 
  — 
  

  

  — 
  3 
  / 
  d 
  u 
  . 
  u 
  8 
  u 
  -v|/ 
  (u, 
  D, 
  t) 
  > 
  . 
  . 
  (2.) 
  

  

  The 
  letter 
  Q,' 
  is 
  here 
  introduced 
  to 
  denote, 
  when 
  negative, 
  that 
  heat 
  which 
  is 
  

   consumed 
  in 
  producing 
  changes 
  of 
  volume 
  and 
  of 
  molecular 
  arrangement, 
  and 
  

   when 
  positive, 
  as 
  in 
  the 
  above 
  equation, 
  the 
  heat 
  which 
  is 
  produced 
  by 
  such 
  

   changes. 
  

  

  The 
  following 
  substitutions 
  have 
  to 
  be 
  made 
  in 
  Equation 
  (1.) 
  of 
  this 
  Section. 
  

  

  For 
  Q 
  is 
  to 
  be 
  substituted 
  its 
  value, 
  according 
  to 
  Equation 
  XII. 
  of 
  the 
  Intro- 
  

   duction 
  ; 
  or 
  abbreviating 
  Gnfib 
  into 
  k 
  : 
  — 
  

  

  Q=2^( 
  T 
  - 
  K 
  ) 
  • 
  • 
  • 
  < 
  3 
  -> 
  

  

  The 
  value 
  of 
  the 
  first 
  integral 
  in 
  Equation 
  (2.) 
  of 
  this 
  Section 
  is 
  

  

  / 
  du 
  . 
  u 
  2 
  -fy 
  (u, 
  D, 
  t)=- 
  

   Jo 
  6 
  

  

  The 
  value 
  of 
  the 
  second 
  integral 
  

  

  — 
  3 
  / 
  du 
  . 
  u 
  8 
  u-^ 
  (w, 
  D, 
  t) 
  

   Jo 
  

  

  remains 
  to 
  be 
  investigated. 
  The 
  first 
  step 
  in 
  this 
  inquiry 
  is 
  given 
  by 
  the 
  condition, 
  

   that 
  whatsoever 
  changes 
  of 
  magnitude 
  a 
  given 
  spherical 
  layer 
  undergoes, 
  the 
  por- 
  

   tion 
  of 
  atmosphere 
  between 
  it 
  and 
  the 
  nucleus 
  is 
  invariable. 
  This 
  condition 
  is 
  

   expressed 
  by 
  the 
  equation 
  

  

  q=(8u~ 
  + 
  8t±. 
  + 
  8-D^\ 
  Tdu 
  . 
  u* 
  + 
  (u,D,t) 
  . 
  . 
  . 
  (4.) 
  

  

  \ 
  du 
  dr 
  dDJjQ 
  

  

  from 
  which 
  it 
  follows 
  that 
  

  

  ^ 
  U= 
  —T-T7 
  — 
  f\ 
  — 
  x 
  l^ 
  T 
  T~ 
  +^D 
  j-n) 
  / 
  du 
  . 
  u 
  2 
  ^(u,D,t) 
  

   tr 
  S> 
  (w, 
  D, 
  t) 
  V 
  dr 
  d\J/jQ 
  

  

  