﻿24 
  R. 
  de 
  Saussure 
  — 
  Graphical 
  Thermodynamtcs. 
  

  

  properties 
  depend 
  essentially 
  upon 
  the 
  choice 
  of 
  the 
  two 
  aux- 
  

   iliary 
  variables 
  a 
  and 
  * 
  ; 
  therefore, 
  they 
  ought 
  to 
  be 
  chosen 
  in 
  

   such 
  a 
  way 
  as 
  to 
  give 
  the 
  best 
  possible 
  graphical 
  representa- 
  

   tion 
  of 
  the 
  cycles 
  of 
  transformation, 
  i. 
  e., 
  in 
  such 
  a 
  way 
  as 
  to 
  

   enable 
  us 
  to 
  determine 
  graphically 
  the 
  greatest 
  possible 
  num- 
  

   ber 
  of 
  the 
  physical 
  elements 
  depending 
  upon 
  the 
  transforma- 
  

   tion, 
  by 
  means 
  of 
  geometrical 
  magnitudes 
  depending 
  only 
  

   upon 
  the 
  form 
  and 
  position 
  of 
  the 
  path 
  described 
  by 
  the 
  body 
  

   in 
  the 
  adopted 
  system 
  of 
  coordinates. 
  Before 
  defining 
  this 
  

   system, 
  let 
  us 
  examine 
  what 
  conditions 
  must 
  be 
  fulfilled 
  by 
  

   the 
  functions 
  X, 
  //, 
  l>, 
  in 
  order 
  that 
  the 
  auxiliary 
  variables 
  a 
  and 
  

   * 
  be 
  respectively 
  the 
  amplitude 
  and 
  the 
  period 
  of 
  the 
  vibratory 
  

   motion. 
  

  

  4. 
  Denoting 
  by 
  m 
  the 
  mass 
  of 
  one 
  of 
  the 
  particles 
  compos- 
  

   ing 
  the 
  substance 
  and 
  by 
  u 
  the 
  mean 
  velocity 
  of 
  the 
  vibratory 
  

   motion, 
  the 
  expression 
  JI»w 
  j 
  is 
  the 
  actual 
  kinetic 
  energy 
  of 
  

   the 
  heat 
  (the 
  sum 
  2 
  1 
  being 
  extended 
  to 
  all 
  the 
  particles). 
  

   Dividing 
  this 
  sum 
  by 
  the 
  mechanical 
  equivalent 
  of 
  heat 
  E, 
  the 
  

   result 
  is 
  equal 
  to 
  the 
  amount 
  of 
  heat 
  actually 
  contained 
  in 
  the 
  

   body. 
  

  

  This 
  amount 
  of 
  heat 
  is 
  proportional 
  to 
  the 
  absolute 
  tempera- 
  

   ture, 
  hence 
  : 
  

  

  i2mu*= 
  KTE 
  (2) 
  

  

  K 
  being 
  a 
  constant. 
  

  

  Denoting 
  by/" 
  the 
  mean 
  value 
  of 
  the 
  force 
  producing 
  the 
  

   vibratory 
  motion, 
  the 
  formulae 
  : 
  

  

  fa 
  = 
  mu* 
  

  

  /=2*>£ 
  (3) 
  

  

  can 
  be 
  established 
  without 
  difficulty, 
  since 
  in 
  all 
  vibratory 
  

   motions 
  of 
  small 
  amplitude, 
  the 
  force 
  producing 
  the 
  vibration 
  

   is 
  proportional 
  to 
  the 
  displacement 
  of 
  the 
  particles. 
  

  

  Combining 
  equations 
  (2) 
  and 
  (3) 
  and 
  noticing 
  that 
  2 
  m 
  = 
  1 
  

   and 
  that 
  the 
  mean 
  velocity 
  u 
  is 
  the 
  same 
  for 
  all 
  the 
  particles 
  of 
  

   the 
  substance, 
  we 
  shall 
  obtain 
  : 
  

  

  -^2 
  -.2 
  

  

  T 
  = 
  — 
  - 
  (4) 
  

  

  KE 
  i 
  2 
  K) 
  

  

  which 
  is 
  the 
  expression 
  of 
  Tin 
  terms 
  of 
  a 
  and 
  i, 
  and 
  is 
  there- 
  

   fore 
  identical 
  to 
  the 
  third 
  of 
  equations 
  (1) 
  ; 
  in 
  other 
  words, 
  

   when 
  the 
  two 
  auxiliary 
  variables 
  a 
  and 
  i 
  denote 
  the 
  amplitude 
  

   and 
  the 
  period 
  of 
  the 
  vibratory 
  motion, 
  the 
  function 
  v 
  is 
  no 
  

   longer 
  arbitrary, 
  and 
  the 
  equation 
  to 
  the 
  thermodynamic 
  sur- 
  

   face 
  is 
  : 
  

  

  (P 
  = 
  A(a,i) 
  

  

  1 
  n* 
  a 
  2 
  ( 
  5 
  ) 
  

  

  I 
  KE 
  € 
  

  

  