EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 91 



cooling the system, there must be a state of uniform temperature 

 for which (regarded as a variation of the original state) 



&?<0 and cty = 0. 



From this we may conclude that for systems of initially uniform 

 temperature condition (2) will not be altered if we limit the variations 

 to such as do not disturb the uniformity of temperature. 



Confining our attention, then, to states of uniform temperature, we 

 have by differentiation of (105) 



Se-tSr] = S\ls + nSt. (112) 



Now there are evidently changes in the system (produced by heating 

 or cooling) for which 



Se-tSti = Q and therefore ^ + 7/^ = 0, (113) 



neither STJ nor St having the value zero. This consideration is 

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



<te-(ty^0, (114) 



and that the condition (111) is equivalent to 



W+qSt^O, (115) 



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



In such cases as we have considered on pages 62-82, in which 

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

 are 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 

 function f, which has been defined for a homogeneous mass on 

 page 87, and which we will here define for any mass of uniform 

 temperature and pressure by the same equation 



=e-tt]+pv. (116) 



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



<#,., ^0. (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 f 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 f for any quantity of the first mass 

 should be equal to the sum of the values of f 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 



