﻿Pressure 
  of 
  Vibrations. 
  345 
  

  

  Boltzmann's 
  theory 
  is 
  founded 
  upon 
  the 
  application 
  of 
  

   Carnot's 
  cycle 
  to 
  the 
  radiation 
  inclosed 
  within 
  movable 
  re- 
  

   flecting 
  walls. 
  If 
  the 
  pressure 
  (p) 
  of 
  a 
  body 
  be 
  regarded 
  

   as 
  a 
  function 
  of 
  the 
  volume 
  v 
  *, 
  and 
  the 
  absolute 
  temperature 
  

   6, 
  the 
  general 
  equation 
  deduced 
  from 
  the 
  second 
  law 
  of 
  ther- 
  

   modynamics 
  is 
  

  

  -^=M, 
  ...... 
  (29) 
  

  

  where 
  M 
  dv 
  represents 
  the 
  heat 
  that 
  must 
  be 
  communicated 
  

   while 
  the 
  volume 
  alters 
  by 
  dv 
  and 
  dd 
  = 
  0. 
  In 
  the 
  application 
  

   of 
  (29) 
  to 
  radiation 
  we 
  have 
  evidently 
  

  

  M 
  = 
  U 
  + 
  p, 
  (30) 
  

  

  where 
  U 
  denotes 
  the 
  density 
  of 
  the 
  energy 
  — 
  a 
  function 
  of 
  6 
  

   only. 
  Hence 
  f 
  

  

  ^= 
  u+ 
  ^ 
  ^ 
  

  

  If 
  further, 
  as 
  for 
  radiation 
  and 
  for 
  aerial 
  vibrations, 
  

  

  *=iU, 
  (32) 
  

  

  it 
  follows 
  at 
  once 
  that 
  

   whence 
  

  

  tflogU 
  = 
  4<nog0. 
  

  

  Ux 
  0\ 
  (33) 
  

  

  the 
  well-known 
  law 
  of 
  Stefan. 
  It 
  may 
  be 
  observed 
  that 
  the 
  

   existence 
  of 
  a 
  pressure 
  is 
  demanded 
  by 
  (31), 
  independently 
  

   of 
  (32). 
  

  

  If 
  we 
  generalize 
  (32) 
  by 
  taking 
  

  

  P=\V, 
  (34) 
  

  

  where 
  n 
  is 
  some 
  numerical 
  quantity, 
  we 
  obtain 
  as 
  the 
  genera- 
  

   lization 
  of 
  (33) 
  

  

  Uoc 
  d 
  n+1 
  (35) 
  

  

  It 
  is 
  an 
  interesting 
  question 
  whether 
  any 
  analogue 
  of 
  the 
  

   second 
  law 
  of 
  thermodynamics 
  can 
  be 
  found 
  in 
  the 
  general 
  

   theory 
  of 
  the 
  pressure 
  of 
  vibrations, 
  whether 
  for 
  example 
  the 
  

   energy 
  of 
  the 
  vibrations 
  of 
  a 
  stretched 
  string 
  is 
  partially 
  

   unavailable 
  in 
  the 
  absence 
  of 
  appliances 
  for 
  distinguishing 
  

   2)hases. 
  It 
  might 
  appear 
  at 
  first 
  sight 
  that 
  the 
  conclusion 
  

   already 
  given, 
  as 
  to 
  the 
  possibility 
  of 
  recovering 
  the 
  whole 
  

   energy 
  by 
  mere 
  retreat 
  of 
  the 
  inclosing 
  ring, 
  was 
  a 
  proof 
  to 
  

  

  * 
  Now 
  with 
  an 
  altered 
  meaning. 
  

  

  f 
  Compare 
  Lorentz, 
  Amsterdam 
  Proceedings, 
  Ap. 
  1901. 
  

  

  