116 J. W. Glbbs — Equilihrium of Heterogeneous Substances. 



We will farther simplify the problem by supposing that the varia- 

 tions of the parts of the energy and entropy which depend upon the 

 surfaces separating heterogeneous masses are so small in comparison 

 with the variations of the parts of the energy and entropy which 

 depend upon the quantities of these masses, that the former may be 

 neglected by the side of the latter; in other words, we will exclude 

 the considerations which belong to the theory of capillarity. 



It will be observed that the siipposition of a rigid and non- 

 conducting envelop enclosing the mass under discussion involves no 

 real loss of genei-ality, for if any mass of matter is in equilibrium, it 

 would also be so, if the whole or any part of it were enclosed in an 

 envelop as supposed ; therefore the conditions of equilibrium for a 

 mass thus enclosed are the general conditions which must always 

 be satisfied in case of equilibrium. As for the other suppositions 

 which have been made, all the circumstances and considerations 

 which are here excluded will afterward be made the subject of 

 special discussion. 



Conditions relating to the Equilibrium between the initially existing 

 Hoinogeneons Partt^ of the given Mass. 



Let us first consider the energy of any homogeneous part of the 

 given mass, and its variation for any j^ossible variation in the com- 

 position and state of this part. (By homogeneous is meant that the 

 part in question is uniform throughout, not only in chemical com- 

 position, but also in physical state.) If we consider the amount and 

 kind of matter in this homogeneous mass as fixed, its energy 5 is a 

 function of its entropy ?/, and its volume v, and the differentials 

 of these quantities are subject to the relation 



ds. ■=. t di] - • p dv ., (11) 



t denoting the (absolute) temperature of the mass, and p its pressure. 

 For t di] is the heat received, and p do the work done, by the mass 

 during its change of state. But if we consider the matter in the 

 mass as variable, and write mj, jn^, . . . m„ for the quantities of the 

 various substances /S'j, /Sg, . . . N„ of which the mass is composed, s 

 will evidently be a function of //, v, m^., ^2, . . . ?>?„, and we shall 

 have for the complete value of the differential of e 



de:=ztdi] — pdv -{- f.i^dm^-\- I.i.,dm2 . . . -|-//„(?ot„, (12) 

 yUj, yWg, . . . //„ denoting the diflferential coefficients of s taken with 

 respect to m,, nio, . . . m„. 



The substances /S',, 62, . . . /S'„, of which we consider the mass 

 composed, must of course be such that the values of the differen- 



