J. W. Gibbs — Kiia'dlbrhmi of IIelero<jeneous Substanceti. 383 



geneous masses, although the mass to which the former quantities 

 rehite is not homogeneous, while in our previous definition of poten- 

 tials, only homogeneous masses were considered. By a natural ex- 

 tension of the term potential, we may call the quantities A ^, A^t 

 etc., \\\Q potentials at the surface of discontimdty. This designation 

 will be fartlier justified by the fact, which will appear hereafter, that 

 the value of these quantities is independent of the thickness of the 

 lamina (M) to which they relate. If we employ our ordinary sym- 

 bols for temperature and potentials, we may write 



6e = t6i]^ fA^ 6)11^ -f /<3 dm^ + etc. (477) 



If we substitue ^ for rr in this equation, the formula will hold 

 true of all variations whether reversible or not ;* for if the variation of 

 energy could have a value less than that of the second member of 

 the equation, there must be variation in the condition of M in which 

 its energy is diminished without change of its entropy or of the 

 quantities of its various components. 



It is important, however, to observe that for any given values of 

 d'v, dm^, Srii^, etc., while there may be possible variations of the 

 nature and state of M for which the value of Se is greater than that 

 of the second member of (477), there must always be possible varia- 

 tions for which the value of 6e is equal to that of the second member. 



* To illustrate the difEerence between variations which are reversible, and those 

 which are not, we may conceive of two entirely different substances meeting in equilib- 

 rium at a mathematical surface without being at all mixed. We may also conceive of 

 them as mixed in a thin film about the surface where they meet, and then the amount 

 of mixture is capable of variation both by increase and by diminution. But when they 

 are a):isolutely unmixed, the amount of mixture can be increased, but is incapable of 

 diminution, and it is then consistent with equilibrium that the value of <)f: (for a varia- 

 tion of the system in which the substances commence to mix) should be greater than 

 the second member of (477). It is not necessary to determine whether precisely such 

 cases actually occur ; but it would not be legitimate to overlook the possible occur- 

 rence of cases in which variations may be possible while the opposite variations are 

 not. 



It will be observed that the sense in whirh the term reversible is here used is en- 

 tirely different from that in which it is frequently used in treatises on thermody- 

 namics, where a process by which a system is brought from a state A to a state B is 

 called reversible, to signify that the system may also be brought from the state B to 

 the state A through the same series of intermediate states taken in the reverse order 

 by means of external agencies of the opposite character. The variation of a system 

 from a state A to a state B (supposed to differ infinitely little from the first) is here 

 called reversible when the system is capable of another state B' which bears the same 

 relation to the state A that A bears to B. 



Trans. Conn. Acad., Vol. III. 49 June, 1877. 



