130 EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 



of these relations, although it may prevent an experimental realization 

 of the phases considered. For the sake of brevity, in the following 

 discussion, phases in the vicinity of the critical phase will generally 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 independently variable components, a pair of 

 coexistent phases will be capable of n independent variations, which 

 may be expressed by the variations of n of the quantities t, p, /x 1} 

 fjL 2 ,...fi n . If we limit these variations by giving to nl of the 

 quantities the constant values which they have for a certain critical 

 phase, we obtain a linear* series of pairs of coexistent phases ter- 

 minated 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 constant 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 second series differing infinitely little from the 

 pair in the first, and vice versa, therefore the second series of coexistent 

 phases must be terminated by a critical phase which differs, but differs 

 infinitely little, from the first. We see, therefore, that if we vary 

 arbitrarily the values of any n 1 of the quantities, t, p, JUL^ ju. 2 , . . . /z n , 

 as determined by a critical phase, we obtain one and only one critical 

 phase for each set of varied values; i.e., a critical phase is capable 

 of n 1 independent variations. 



The quantities t, p, JJL V /m. 2 , . . . JUL U have the same values in two 

 coexistent phases, but the ratios of the quantities r], v } m v m z , . . . m n 

 are in general different in the two phases. Or, if for convenience we 

 compare equal volumes of the two phases (which involves no loss of 

 generality), the quantities q, m v m 2 , ... m n will in general have dif- 

 ferent values in two coexistent phases. Applying this to coexistent 

 phases indefinitely near to a critical phase, we see that in the 

 immediate vicinity of a critical phase, if the values of n of the 

 quantities t, p, fi lt yM 2 , ... /x n are regarded as constant (as well as v), 

 the variations of either of the others will be infinitely small compared 

 with the variations of the quantities 77, m v m 2 , . . . m n . This condition, 

 which we may write in the form 



=- (200 > 



Vt w ,".Mn-i 



characterizes, as we have seen on page 114, the limits which divide 

 stable from unstable phases in respect to continuous changes. 



In fact, if we give to the quantities t, JUL V JUL Z , ... fi n - 1 constant values 



* This term is used to characterize a series having a single degree of extension. 



