I 88 J. IK Gibbs — Equ'dibruuii. of Heterogeneous Substances. 



pivssuro. For if, ;it iiiiy given tcniixTutuiv and pressure, two liquids 

 nre ca})iil)le of forming a stable mixture in any ratio in ^ : m^ less than 

 rt, and in any greater than A, n and h being the values of that ratio 

 for two coexistent ))hases, while either can form a stable mixture with 

 a third licjuid in all jtroportions, and any small quantities of the iirst 

 and second can unite at once with a great quantity of the third to 

 form a stable mixture, it may easily be seen that two coexistent mix- 

 tures of the three liquids may be varied in composition, the tempera- 

 ture and pressure remaining the same, from initial phases in each of 

 which the (piantity of the third liquid is nothing, to a terminal phase 

 in whicli the distinction of the two phases vanishes. 



In general, we may define a critical phase as one at which the dis- 

 tinction between coexistent i>hases vanishes. We may suppose the 

 coexistent phases to be stable in respect to continuous changes, for 

 although I'elations in some icspects analogous might be imagined to 

 hold true in regard to ])hases which are unstable in respect to con- 

 tinuous changes, the discussion of siudi cases would be devoid of 

 interest. But if the coexistent jthases and the critical phase are 

 unstable only in respect to the possible formation of phases entirely 

 ditferent from the critical and adjacent phases, the liability to such 

 changes will in no respect affect the relations between the critical and 

 adjacent jdiases, and need not be considered in a theoretical discussion 

 of these relations, although it may prevent an experimental realiza- 

 tion of the phases considered. For the sake of brevity, in the follow- 

 ing discussion, ])hases in tlu^ vicinity of the critical phase will gen- 

 erally be called stable, if they are unstable only in respect to the 

 formation of phases entirely different from any in the vicinity of the 

 critical phase. 



Let us first consider the number of independent variations of which 

 a critical phase (while remaining such) is capable. If we denote by 

 n the number of indejiendently variable components, a pair of coexis- 

 tent phases will be capable of n independent variations, which may be 

 expressed by the variations of ti of the quantities t, p, //^, //^, ...//„. 

 If we limit these variations by giving to n — 1 of the quantities the 

 constant values which they have for a certain critical phase, we 

 obtain a linear* series of pairs of coexistent phases terminated by the 

 critical phase. If we now vary infinitesimally the values of these 

 n — l quantities, we shall have for the new set of values considered con- 

 stant a new linear series of pairs of coexistent phases. Now for every 

 pair of phases in the first series, there must be pairs of phases in the 



* This tonn is used to cliaracterize a series having a single degree of extension. 



