J. W. Gibbs — Equilibrium, of Heterogeneous Substances. 193 



than n of these quantities or functions of tlie same together with cer- 

 tain of the quantities ?;, u, wij, Wj, . . . m„, 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 i?„+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^^\ and m = 2. If a mass of invariable composi- 

 tion is in a critical state, we may keep its volume constant, and 

 destroy its homogeneity 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 homogeneity 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 con- 

 stant, the mass will remain homogeneous, however the proportion of 

 the components 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 state- 

 ments 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. Duolaux, " Sur la separation des liquides melanges, etc." will be 

 found in Comptes Bendus, vol. Ixxxi. (1875), p. 815. 



Trans. Conn. Acad., Vol. III. 26 February, 1876. 



