EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 247 



from that of which the stability is in question. These varied states 

 of the system are not in general states of equilibrium, and the 

 relations expressed by the fundamental equations may not hold true 

 of them. More than this, if we attempt to describe a varied state of 

 the system by varied values of the quantities which describe the 

 initial state, if these varied values are such as are inconsistent with 

 equilibrium, they may fail to determine with precision any state of 

 the system. Thus, when the phases of two contiguous homogeneous 

 masses are specified, if these phases are such as satisfy all the 

 conditions of equilibrium, the nature of the surface of discontinuity 

 (if without additional components) is entirely determined ; but if the 

 phases do not satisfy all the conditions of equilibrium, the nature of 

 the surface of discontinuity is not only undetermined, but incapable 

 of determination by specified values of such quantities as we have 

 employed to express the nature of surfaces of discontinuity in 

 equilibrium. For example, if the temperatures in contiguous homo-. 

 geneous masses are different, we cannot specify the thermal state 

 of the surface of discontinuity by assigning to it any particular 

 temperature. It would be necessary to give the law by which the 

 temperature passes over from one value to the other. And if this 

 were given, we could make no use of it in the determination of other 

 quantities, unless the rate of change of the temperature were so 

 gradual that at every point we could regard the thermodynamic state 

 as unaffected by the change of temperature in its vicinity. It is true 

 that we are also ignorant in respect to surfaces of discontinuity in 

 equilibrium of the law of change of those quantities which are 

 different in the two phases in contact, such as the densities of the 

 components, but this, although unknown to us, is entirely determined 

 by the nature of the phases in contact, so that no vagueness is 

 occasioned in the definition of any of the quantities which we have 

 occasion to use with reference to such surfaces of discontinuity. 



It may be observed that we have established certain differential 

 equations, especially (497), in which only the initial state is necessarily 

 one of equilibrium. Such equations may be regarded as establishing 

 certain properties of states bordering upon those of equilibrium. But 

 these are properties which hold true only when we disregard quantities 

 proportional to the square of those which express the degree of 

 variation of the system from equilibrium. Such equations are there- 

 fore sufficient for the determination of the conditions of equilibrium, 

 but not sufficient for the determination of the conditions of stability. 



We may, however, use the following method to decide the question 

 of stability in such a case as has been described. 



Beside the real system of which the stability is in question, it will 

 be convenient to conceive of another system, to which we shall 



