EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 131 



/yvi 



determined by a pair of coexistent phases, and to - - a series of 



values increasing from the less to the greater of the values which it 

 has in these coexistent phases, we determine a linear series of phases 

 connecting the coexistent phases, in some part of which juL n since it 

 has the same value in the two coexistent phases, but not a uniform 

 value throughout the series (for if it had, which is theoretically im- 

 probable, all these phases would be coexistent) must be a decreasing 



vn 



function of , or of m n , if v also is supposed constant. Therefore, 



the series must contain phases which are unstable in respect to con- 

 tinuous changes. (See page 111.) And as such a pair of coexistent 

 phases may be taken indefinitely near to any critical phase, the 

 unstable phases (with respect to continuous changes) must approach 

 indefinitely near to this phase. 



Critical phases have similar properties 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 

 quantities t, p, fa, fa, ... /u. n the 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 

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

 is varied in such a manner that n of the quantities t, p, fa, fa, ... JUL U 

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

 and to discontinuous changes. Therefore /m n is an increasing function 

 of m n when t, v, fa, fa, ... JUL U - 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 : 



=0, (201) 



t, V, Ml> Mn-1 



(202) 



Mn-l 



If the sign of equality holds in the last condition, additional conditions, 

 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 of n 1 inde- 

 pendent variations. 



