﻿MECHANICAL 
  ACTION 
  OF 
  HEAT. 
  159 
  

  

  compressing 
  a 
  body 
  appears 
  in 
  the 
  form 
  of 
  heat. 
  More 
  or 
  less 
  power 
  may 
  be 
  

   consumed 
  or 
  developed 
  by 
  changes 
  of 
  molecular 
  arrangement, 
  or 
  of 
  the 
  internal 
  

   distribution 
  of 
  the 
  density 
  of 
  the 
  atomic 
  atmospheres 
  ; 
  and 
  changes 
  of 
  molecular 
  

   arrangement 
  or 
  distribution 
  may 
  develope 
  or 
  consume 
  heat, 
  independently 
  of 
  

   changes 
  of 
  volume. 
  

  

  (6.) 
  We 
  shall 
  now 
  investigate, 
  according 
  to 
  the 
  hypothesis 
  of 
  molecular 
  

   vortices, 
  the 
  amount 
  of 
  heat 
  produced 
  by 
  an 
  indefinitely 
  small 
  compression 
  of 
  one 
  

   atom 
  of 
  a 
  body 
  in 
  that 
  state 
  of 
  perfect 
  fluidity 
  which 
  admits 
  of 
  the 
  bounding 
  

   surface 
  of 
  the 
  atom 
  being 
  treated 
  as 
  if 
  it 
  were 
  spherical 
  : 
  its 
  radius 
  being 
  denoted 
  

   by 
  R, 
  and 
  the 
  radius 
  of 
  any 
  internal 
  spherical 
  layer 
  of 
  the 
  atmosphere 
  by 
  multi- 
  

   plying 
  R 
  by 
  a 
  fraction 
  u. 
  

  

  I 
  shall 
  denote 
  by 
  the 
  ordinary 
  symbol 
  of 
  differentiation 
  d, 
  such 
  changes 
  as 
  

   depend 
  on 
  the 
  various 
  positions 
  of 
  portions 
  of 
  the 
  atomic 
  atmosphere 
  relatively 
  

   to 
  each 
  other, 
  when 
  changes 
  of 
  volume 
  and 
  temperature 
  are 
  not 
  taken 
  into 
  con- 
  

   sideration 
  ; 
  while 
  by 
  the 
  symbol 
  8 
  of 
  the 
  calculus 
  of 
  variations, 
  I 
  shall 
  represent 
  

   such 
  changes 
  as 
  arise 
  from 
  the 
  variations 
  of 
  volume 
  and 
  temperature. 
  

  

  Let 
  us 
  consider 
  the 
  case 
  of 
  an 
  indefinitely 
  thin 
  spherical 
  layer 
  of 
  the 
  atomic 
  

   atmosphere, 
  whose 
  distance 
  from 
  the 
  nucleus 
  is 
  Rw, 
  its 
  thickness 
  Rdu, 
  its 
  area 
  

  

  4 
  7T 
  R 
  2 
  m 
  2 
  , 
  and 
  its 
  density 
  ^ 
  D 
  4* 
  (u, 
  D, 
  t). 
  

  

  The 
  weight, 
  then, 
  of 
  this 
  layer 
  is 
  

  

  4 
  7T 
  R 
  3 
  -^ 
  D 
  « 
  2 
  ^ 
  ( 
  M 
  , 
  D 
  ; 
  t) 
  <^. 
  

  

  Its 
  velocity 
  of 
  oscillation 
  is 
  v, 
  and 
  having, 
  in 
  virtue 
  of 
  that 
  velocity, 
  a 
  mean 
  cen- 
  

   trifugal 
  force, 
  as 
  explained 
  in 
  the 
  Introduction 
  (Equation 
  V.), 
  equal 
  to 
  

  

  its 
  weight 
  x 
  (-^- 
  = 
  -M 
  ) 
  

  

  it 
  is 
  kept 
  in 
  equilibrio 
  by 
  an 
  equal 
  and 
  opposite 
  centripetal 
  force, 
  arising 
  from 
  

   attraction 
  and 
  elastic 
  pressure, 
  which 
  is 
  consequently 
  represented 
  by 
  

  

  4 
  7T 
  R 
  2 
  ^ 
  —. 
  ■ 
  D 
  u 
  ^ 
  («, 
  D, 
  t) 
  du 
  

  

  NL 
  g 
  k 
  v 
  

  

  = 
  8irR 
  2 
  iQD«4(«,D 
  ) 
  T)(/«. 
  

  

  Let 
  the 
  mean 
  density 
  of 
  the 
  atom 
  now 
  be 
  increased 
  by 
  the 
  indefinitely 
  small 
  

   quantity 
  8 
  J). 
  Then 
  the 
  layer 
  will 
  approach 
  the 
  nucleus 
  through 
  the 
  distance 
  

   -5(Rm)=-m^R-R^m, 
  and 
  being 
  acted 
  upon 
  through 
  that 
  distance 
  by 
  the 
  cen- 
  

   tripetal 
  force 
  already 
  stated, 
  the 
  vis 
  viva 
  of 
  oscillation 
  will 
  be 
  increased 
  by 
  a 
  

   quantity 
  corresponding 
  to 
  the 
  mechanical 
  power 
  (that 
  is 
  to 
  say, 
  the 
  heat), 
  repre- 
  

   sented 
  by 
  the 
  product 
  of 
  that 
  distance 
  by 
  that 
  force, 
  or 
  by 
  

  

  -8 
  7T 
  R 
  2 
  -£-= 
  QD 
  u 
  4, 
  (u, 
  D,7)duxd 
  (R 
  ») 
  

  

  fC 
  J.YJ. 
  

  

  VOL. 
  XX. 
  PART 
  I. 
  2 
  U 
  

  

  