190 J. W. Gibbs — Equilibrmin of Heterogeneous Substances. 



Critical phases have similar pi'operties with reference to stability 

 as determined with regard to discontinuous changes. For as every 

 stable phase which has a coexistent phase lies upon the limit which 

 separates stable from unstable phases, the same must be true of any 

 stable critical phase. (The same may be said of critical phases which 

 are unstable in regard to discontinuous changes if we leave out of 

 account the liability to the particular kind of discontinuous change 

 in respect to which the critical phase is unstable.) 



The linear series of phases determined by giving to n of the quanti- 

 ties t,p,Mi-'M2i ' • • /'" ^^^ constant values which they have in any 

 pair of coexistent phases consists of unstable phases in the part 

 between the coexistent phases, but in the part beyond these phases in 

 eithei" direction it consists of stable phases. Hence, if a critical phase 

 is varied in such a manner ihntn of the quantities t,p, /.i^, yUg, . . . /v„ 

 remain constant, it will remain stable in respect both to continuous and 

 to discontinuous changes. Therefore, yu„.is an increasing function of 

 m„ when t, v, j^i^, /.I2, • • • /'n-i have constant values determined by 

 any critical phase. But as equation (200) holds true at the critical 

 phase, the following conditions must also hold true at that phase : 



fd^/n„\ 



= 0, (201) 





\d}n„^)t, V, 



^0. (202) 



If the sign of equality holds in the last condition, additional condi- 

 tions, concerning the differential coefficients of higher orders, must be 

 satisfied. 



Equations (200) and (201) may in general be called the equations 

 of critical phases. It is evident that there are only two independent 

 equations of this character, as a critical phase is capable oi n—l inde- 

 pendent variations. 



We are not, however, absolutely certain that equation (200) will 

 always be satisfied by a critical phase. For it is possible that the 

 denominator in the fraction may vanish as well as the numerator for 

 an infinitesimal change of phase in which the quantities indicated 

 are constant. In such a case, we may suppose the subscript n to 

 refer to some different component substance, or use another differen- 

 tial coefficient of the same general form (such as are described on 

 page 171 as characterizing the limits of stability in respect to con- 

 tinuous changes), making the corresponding changes in (201) and 

 (202). We may be certain that some of the formula^ thus formed 

 will not fail. But for a perfectly rigorous method there is an ad van- 



