ABSTKACT BY THE AUTHOR. 361 



existence in virtue of properties which prevent the commencement of 

 discontinuous changes. But a phase which is unstable with respect to 

 continuous changes is evidently incapable of permanent existence on a 

 large scale except in consequence of passive resistances to change. 

 To obtain the conditions of stability with respect to continuous 

 changes, we have only to limit the application of the variables in (14) 

 to phases adjacent to the given phase. We obtain results of the 

 following nature. 



The stability of any phase with respect to continuous changes 

 depends upon the same conditions with respect to the second and 

 higher differential coefficients of the density of energy regarded as 

 a function of the density of entropy and the densities of the several 

 components, which would make the density of energy a minimum, 

 if the necessary conditions with respect to the first differential 

 coefficients were fulfilled. 



Again, it is necessary and sufficient for the stability with respect 

 to continuous changes of all the phases within any given limits,~that 

 within those limits the same conditions should be fulfilled with 

 respect to the second and higher differential coefficients of the 

 pressure regarded as a function of the temperature and the several 

 potentials, which would make the pressure a minimum, if the 

 necessary conditions with respect to the first differential coefficients 

 were fulfilled. 



The equation of the limits of stability with respect to continuous 

 changes may be written 



=0 , or =00, (15) 



where y n denotes the density of the component specified or m n -r-v. 

 It is in general immaterial to what component the suffix n is regarded 

 as relating. 



Critical phases. The variations of two coexistent phases are 

 sometimes limited by the vanishing of the difference between them. 

 Phases at which this occurs are called critical phases. A critical 

 phase, like any other, is capable of Ti-fl independent variations, 

 n denoting the number of independently variable components. But 

 when subject to the condition of remaining a critical phase, it is 

 capable of only n 1 independent variations. There are therefore 

 two independent equations which characterize critical phases. These 

 may be written 



=Q 



It will be observed that the first of these equations is identical with 

 the equation of the limit of stability with respect to continuous 



