J. W. Gibbs — Equilibrium of Heterogeneous Substances. 155 



When n=L 1, 



{m" v' ~ m' v") dp = {m" if - m //") dt, (130) 



or, if we make ni' =. 1 and m" =. 1, we liave the usual formula 



dt v'-v" t{v"-v'y ^ ^ 



in which Q denotes the heat absorbed by a unit of the substance in 

 passing from one state to the other without change of temperature or 

 pressure. 



Co7icerning Cases in which the Number of Coexistent Phases is less 



than /i-J- 1. 



When M> 1, if the quantities of all the components /S'j, /Sg, . . . S„ 

 are proportional in two coexistent phases, the two equations of the 

 form of (127) and (128) relating to these phases will be sufficient 

 for the elimination of the variations of all the potentials. In fact, 

 the condition of the coexistence of the two phases together with the 

 condition of the equality of the n — 1 ratios of «*/, m^ ^ , . . m„' 

 with the n — \ ratios of m^\ ^'^-z" ■> • • • ''*"' ^^ sufficient to detei'mine 

 /> as a function of t if the fundamental equation is known for each of 

 the phases. The ditferential equation in this case may be expressed 

 in the form of (130), m' and m" denoting either the quantities of any 

 one of the components or the total quantities of matter in the bodies 

 to which they relate. Equation (131) will also hold true in this case, 

 if the total quantity of matter in each of the bodies is unity. But 

 this case differs from the preceding in that the matter which absorbs 

 the heat Q in passing from one stat j to another, and to which the other 

 letters in the formula relate, alt-iough the same in quantity, is not in 

 general the same in kind at different temperatures and pressures. 

 Yet the case wall often occur that one of the phases is essentially 

 invariable in composition, especially when it is a crystalline body, 

 and in this case the matter to which the letters in (131) relate will 

 not vary with the temperature and pressure. 



When n = 2, two coexistent phases are capable, when the temper- 

 ature is constant, of a single variation in phase. But as (130) will 

 hold true in this case when m/ : m^' : : m^" : m^", it follows that for 

 constant temperature the pressure is in general a maximum or a min- 

 imum when the composition of the two phases is identical. In like 

 manner, the temperature of the two coexistent phases is in general a 

 maximum or a minimum, for constant pi-essure, when the composition 

 of the two phases is identical. Hence, the series of simultaneous 

 values of t and p for which the composition of two coexistent phases 



