﻿P. 
  de 
  Saussure 
  — 
  Graphical 
  Thermodynamics. 
  23 
  

  

  and 
  the 
  physical 
  state 
  can 
  be 
  represented 
  on 
  a 
  piece 
  of 
  paper 
  

   by 
  the 
  point 
  corresponding 
  to 
  these 
  coordinates. 
  In 
  this 
  case, 
  

   the 
  piece 
  of 
  paper 
  itself 
  or 
  a 
  part 
  of 
  it, 
  is 
  the 
  representative 
  

   locus 
  of 
  the 
  different 
  states 
  under 
  which 
  the 
  substance 
  can 
  

   exist. 
  

  

  It 
  follows 
  that 
  each 
  point 
  of 
  the 
  thermodynamic 
  surface 
  

   F(P, 
  Y, 
  T) 
  = 
  corresponds 
  to 
  a 
  point 
  on 
  the 
  sheet 
  of 
  paper, 
  

   and 
  conversely. 
  If 
  the 
  variables 
  P, 
  Y 
  and 
  T 
  vary 
  continu- 
  

   ously, 
  the 
  variables 
  a 
  and 
  i 
  shall 
  also 
  vary 
  continuously, 
  since 
  

   the 
  variation 
  of 
  the 
  state 
  of 
  the 
  body 
  is 
  itself 
  continuous 
  ; 
  so 
  

   that 
  the 
  coordinates 
  P, 
  F, 
  T 
  are 
  continuous 
  functions 
  of 
  the 
  

   coordinates 
  a 
  and 
  i. 
  

  

  P 
  = 
  \( 
  a 
  , 
  i) 
  

  

  V 
  = 
  /l(a 
  f 
  t) 
  (1) 
  

  

  T 
  = 
  v(a, 
  i) 
  

  

  These 
  three 
  equations 
  can 
  be 
  regarded 
  as 
  the 
  general 
  equation 
  

   to 
  the 
  thermodynamic 
  surface 
  in 
  terms 
  of 
  two 
  auxiliary 
  vari- 
  

   ables 
  a 
  and 
  i 
  ; 
  hence, 
  by 
  eliminating 
  a 
  and 
  i 
  between 
  them, 
  

   the 
  result 
  must 
  be 
  : 
  F(P, 
  Y, 
  T) 
  = 
  0. 
  

  

  The 
  variables 
  a 
  and 
  i 
  can 
  be 
  considered 
  as 
  the 
  coordinates 
  

   of 
  any 
  point 
  on 
  the 
  thermodynamic 
  surface. 
  Any 
  relation 
  

   between 
  a 
  and 
  i 
  represents 
  a 
  curve 
  traced 
  on 
  this 
  surface, 
  i. 
  e., 
  

   a 
  cycle 
  of 
  transformations 
  undergone 
  by 
  the 
  substance. 
  We 
  

   have 
  just 
  seen 
  that 
  the 
  functions 
  X, 
  //, 
  u 
  must 
  be 
  such 
  as 
  to 
  

   lead 
  to 
  the 
  relation 
  F(P, 
  Y, 
  T) 
  = 
  by 
  eliminating 
  a 
  and 
  i 
  

   between 
  equations 
  (1) 
  ; 
  but 
  as 
  long 
  as 
  these 
  functions 
  are 
  sub- 
  

   mitted 
  only 
  to 
  this 
  condition, 
  the 
  variables 
  a 
  and 
  i 
  are 
  still 
  

   arbitrary 
  variables 
  and 
  do 
  not 
  necessarily 
  denote 
  the 
  amplitude 
  

   and 
  the 
  period 
  of 
  the 
  vibratory 
  motion, 
  since 
  there 
  are 
  an 
  

   infinite 
  number 
  of 
  ways 
  of 
  representing 
  the 
  same 
  surface 
  by 
  

   means 
  of 
  two 
  auxiliary 
  variables. 
  For 
  instance, 
  the 
  equation 
  

   of 
  the 
  body 
  : 
  T 
  =f(P, 
  Y) 
  can 
  be 
  put 
  under 
  the 
  form 
  : 
  

  

  F 
  = 
  u 
  

  

  V=v 
  

  

  T 
  =f(u, 
  v) 
  

  

  u 
  and 
  v 
  being 
  the 
  two 
  auxiliary 
  coordinates 
  chosen 
  to 
  repre- 
  

   sent 
  graphically 
  the 
  cycles 
  of 
  transformations. 
  

  

  These 
  coordinates 
  u 
  and 
  v 
  being 
  equal 
  to 
  P 
  and 
  ^respec- 
  

   tively, 
  the 
  graphical 
  representation 
  thus 
  obtained 
  would 
  be 
  

   the 
  same 
  as 
  the 
  one 
  first 
  introduced 
  in 
  thermodynamics 
  by 
  

   Clapeyron, 
  and 
  would 
  have 
  the 
  same 
  property, 
  i. 
  e., 
  the 
  area 
  

   lying 
  between 
  the 
  axis 
  of 
  V, 
  two 
  ordinates 
  and 
  the 
  path 
  

   described 
  by 
  the 
  body 
  would 
  be 
  equal 
  to 
  the 
  external 
  work. 
  

  

  If 
  other 
  coordinates 
  are 
  chosen, 
  the 
  properties 
  of 
  the 
  graph- 
  

   ical 
  representation 
  will 
  change, 
  for 
  it 
  is 
  evident 
  that 
  these 
  

  

  