J. W. Gibbs — Equilibrium of Heterogeneous Substances. 147 



neither S/; nor dt having the value zero. This consideration is suffi- 

 cient to show that the condition (2) is equivalent to 



de — tdf/^0. (114) 



and that the condition (111) is equivalent to 



Sif^-^}/6t^0 . (115) 



and by (112) the two last conditions are equivalent. 



In such cases as we have considered on pages 115-137, in which 

 the form and position of the masses of which the system is composed 

 is immaterial, uniformity of temperature and pressure are always 

 necessary for equilibrium, and the remaining conditions, when these 

 are satisfied, may be conveniently expressed by means of the func- 

 tion ?, which has been defined for a homogeneous mass on page 142, 

 and which we will here define for any mass of uniform temperature 

 and pressure by the same equation 



t,^£ — ttj-\-pv. (Ii6) 



For such a mass, the condition of (internal) equilibrium is 



m,,^o. (117) 



That this condition is equivalent to (2) will easily appear from con- 

 siderations like those used in respect to (111). 



Hence, it is necessary for the equilibrium of two contiguous masses 

 identical in composition that the values of C as determined for equal 

 quantities of the two masses should be equal. Or, when one of three 

 contiguous masses can be formed out of the other two, it is necessary 

 for equilibrium that the value of C for any quantity of the first mass 

 should be equal to the sum of the values of t. for such quantities of the 

 second and third masses as together contain the same matter. Thus, 

 for the equilibrium of a solution composed of a parts of water and b 

 parts of a salt which is in contact with vapor of water and crystals of 

 the salt, it is necessary that the value of t, for the quantity a-\-b oi the 

 solution should be equal to the sum of the values of C for the quanti- 

 ties a of the vapor and b of the salt. Similar propositions will hold 

 true in more complicated cases. The reader will easily deduce these 

 conditions from the particular conditions of equilibrium given on 

 page 128. 



In like manner we may extend the definition of x to any mass or 



combination of masses in which the pressure is everywhere the same, 



using e for the energy and v for the volume of the whole and setting 



as before 



X=e-\-pv. (118) 



