382 J. TF! Gibhs — Equilihrium of Heterogeneous Substances. 



beyond the bounding surface the phase is identical with that within 

 the surface. This will evidently be allowable, if it does not affect 

 tlie total amount of energy. To show that it does not affect this 

 quantity, Ave have only to observe that, if the energy of the mass on 

 one side of a surface where there is an infinitesimal discontinuity of 

 phase is greater as determined l)y this rule than if determined by 

 any other (suitable) rule, the energy of the mass on the other side 

 must be less by the same amount when determined by the first rule 

 than when determined by the second, since the discontinuity I'elative 

 to the second mass is equal but opposite in character to the discon- 

 tinuity relative to the first. 



If the entropy of the mass which occupies any one of the spaces 

 considered is not in the nature of things determined without refer- 

 ence to the surrounding masses, we may suppose a similar method to 

 be applied to the estimation of entropy. 



With this understanding, let us return to the consideration of the 

 equilibrium of the three masses M, M', and W. We shall suppose 

 that there are no limitations to the possible variations of the system 

 due to any want of perfect mobility of the components by means of 

 which we express the composition of the masses, and that these com- 

 ponents are independent, i. e,, that no one of them can be formed out 

 of the others. 



With regard to the mass M, which includes the surface of discon- 

 tinuity, it is necessary for its internal equilibrium tliat when its 

 boundaries are considered constant, and when we consider only 

 reversible variations (i. e., those of which the opposite are also 

 possible), the variation of its energy should vanish with the varia- 

 tions of its entropy and of the quantities of its various components. 

 For changes within this mass will not affect the energy or the entropy 

 of the surrounding masses (when these quantities are estimated on 

 the principle whicli we have adopted), and it may therefore be 

 treated as an isolated system. For fixed boundaries of the mass M, 

 and for reversible variations, we may therefoi'e write 



de = A^^di] -\- A^ Sm^ + A^ 6m ^ + etc, (476) 



where yl„, A^, A2, etc., are quantities determined l)y the initial 

 (ixnvaried) condition of the system. It is evident that A^ is the 

 teni])eratiire of the lamelliform mass to which tlie equation relates, 

 or tlie temperature at the surface of discontinuity. By comparison 

 of this equation with (12) it will be seen that the definition of yj ,, 

 ^^2, etc, is entirely analogous to that of the potentials in homo- 



