J. W. Gibbs — Equilibr turn of Heterogeneous Substances. 411 



necessary to give the law by which tlie 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 unaflTected by the change 

 of temperature in its vicinity. It is true that we are also ignorant in 

 respect to surfaces of discontinuity iti equiUbriimi 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 ol discontinuity. 



It may be observed that we have established certain ditterential 

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

 sarily one of equilibrium. Such equations may be regarded as estab- 

 lishing certain properties of states bordering upon those of equilib- 

 rium, J5ut these are properties which hold true only when we dis- 

 regard quantities proportional to the square of those which express 

 the degree of variation of the system from equilibrium. Such equa- 

 tions are therefore sufficient for the determination of the conditions of 

 equilibrium, but not sufficient for the determination of the conditions 

 of stability 



We may, howevei", 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 attri- 

 bute in its initial state the same homogeneous masses and surfaces of 

 discontinuity which belong to the real system. We shall also sup- 

 pose that the homogeneous masses and surfaces of discontinuity of 

 this system, which we may call the imaginary system, have the same 

 fundamental equations as those of the real system. But the imagin- 

 ary system is to differ from the real in that the variations of its state 

 are limited to such as do not violate the conditions of equilibrium 

 relating to temperature and the potentials, and that the fundamental 

 equations of the surfaces of discontinuity hold true for these varied 

 states, although the condition of equilibrium expressed by equation 

 (500) may not be satisfied. 



Before proceeding farther, we must decide whethei- we are to 

 examine the question of stability under the condition of a constant 

 external temperature, or under the condition of no transmission of 



