134 EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 



there will correspond three different phases, of which one is stable 

 with respect both to continuous and to discontinuous changes, another 

 is stable with respect to the former and unstable with respect to the 

 latter, and the third is unstable with respect to both. 



In general, however, if n of the quantities p, t, //j, // 2 , ... /z n , or n 

 arbitrary functions of these quantities, have the same constant values 

 as at a critical phase, the linear series of phases thus determined will 

 be stable, in the vicinity of the critical phase. But if less than n of 

 these quantities or functions of the same together with certain of the 

 quantities r\, v, m l} m 2 ,...m n , or arbitrary functions of the latter 

 quantities, have the same values as at a critical phase, so as to 

 determine a linear series of phases, the differential of R n +i in such a 

 series of phases will not in general vanish at the critical phase, so that 

 in general a part of the series will be unstable. 



We may illustrate these relations by considering separately the cases 

 in which n = 1 and n = 2. If a mass of invariable composition is in a 

 critical state, we may keep its volume constant, and destroy its homo- 

 geneity by changing its entropy (i.e., by adding or subtracting heat 

 probably the latter), or we may keep its entropy constant and destroy 

 its homogeneity by changing its volume ; but if we keep its pressure 

 constant we cannot destroy its homogeneity by any thermal action, 

 nor if we keep its temperature constant can we destroy its homo- 

 geneity by any mechanical action. 



When a mass having two independently variable components is in 

 a critical phase, and either its volume or its pressure is maintained 

 constant, its homogeneity may be destroyed by a change of entropy 

 or temperature. Or, if either its entropy or its temperature 'is main- 

 tained constant, its homogeneity may be destroyed by a change of 

 volume or pressure. In both these cases it is supposed that the 

 quantities of the components remain unchanged. But if we suppose 

 both the temperature and the pressure to be maintained constant, the 

 mass will remain homogeneous, however the proportion of the com- 

 ponents be changed. Or, if a mass consists of two coexistent phases, 

 one of which is a critical phase having two independently variable 

 components, and either the temperature or the pressure of the mass is 

 maintained constant, it will not be possible by mechanical or thermal 

 means, or by changing the quantities of the components, to cause the 

 critical phase to change into a pair of coexistent phases, so as to give 

 three coexistent phases in the whole mass. The statements of this 

 paragraph and of the preceding have reference only to infinitesimal 

 changes.* 



* A brief abstract (which came to the author's notice after the above was in type) of a 

 memoir by M. Duclaux, " Sur la separation des liquides melanges, etc." will be found in 

 Comptes Rendus, vol. Ixxxi. (1875), p. 815. 



