J. W. Gihhs — Equilihrium of Heterogeneous Substances. 415 



where -^— , and ~, are to be determined from tlie form of tlie surtace 

 dv dv 



of tension by piirely geometrical considerations, and the other dift'er- 

 ential coefficients are to be determined from the fundamental equa- 

 tions of the homogeneous masses and the surface of discontinuity. 

 Condition (540) may be easily deduced from this as a particular case. 

 The condition of stability with reference to motion of surfaces of 

 discontinuity admits of a very simple expression when we can ti'eat 

 the temperature and potentials as constant. This will be the case 

 when one or more of the homogeneous masses, containing together 

 all the component substances, may be considered as indefinitely large, 

 the surfaces of discontinuity being finite. For if we write 2/i8 for 

 the sum of the variations of the energies of the several homogeneous 

 masses, and 2Ae^ for the sum of the variations of the energies of the 

 several surfaces of discontinuity, the condition of stability may be 

 WTitten 



2Je + :S^z/6S>o, (548) 



the total entropy and the total quantities of the several components 

 l>eing constant. The variations to be considered are infinitesimal, 

 but the character J signifies, as elsewhere in this paper, that the ex- 

 pression is to be interpreted without neglect of infinitesimals of the 

 higher orders. Since the temperature and potentials are sensibly con- 

 stant, the same will be true of the pressures and surface-tensions, and 

 by integration of (86) and (501) we may obtain for any homogeneous 

 mass 



Ae ^zt J?/ ^ p Jv -[- /-<, Arn^ -\- jn^ ^m^ -\- etc., 



and for any surface of discontinuity 



At^ = t Ajf J^ (J As-\- fi^ Am.\ + //| Am^ + etc. 



These equations will hold true of finite differences, when t, /?, c, ja 

 /<2? etc. ave constant, and will therefore hold true of infinitesimal dif- 

 ferences, under the same limitations, without neglect of the infinitesi- 

 mals of the higher orders. By substitution of these values, the condi- 

 tion of stability will reduce to the form 



- 2{2)Av) + 2{()As) > 0, 



or 2{2)Av) — 2{(jAs) < 0. (549) 



That is, the sum of the products of the volumes of the masses bv 

 their pi-essures diminished by the sum of the products of the areas of 

 the surfaces of discontinuity by their tensions must be a maximum. 

 This is a purely geometrical condition, since the pressures and ten- 

 Trans. Conn. Acad., Vol. III. 53 Nov., ISTT. 



