222 EQUILIBKIUM OF HETEROGENEOUS SUBSTANCES. 



If we substitute 2: for = 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 

 Sri, Sm ly Sm 2 , 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 Se is equal to that of the second member. 

 It will be convenient to have a notation which will enable us to 

 express this by an equation. Let be denote the smallest value (i.e., the 

 value nearest to oo ) of Se consistent with given values of the other 



variations, then 



be = tSr)-^-iuL l Sm 1 + fi 2 8m z + etc. (478) 



For the internal equilibrium of the whole mass which consists of 

 the parts M, M', M", it is necessary that 



&+&' + &"^0 (479) 



for all variations which do not affect the enclosing surface or the 

 total entropy or the total quantity of any of the various components. 

 If we also regard the surfaces separating M, M', and M" as invariable, 

 we may derive from this condition, by equations (478) and (12), the 

 following as a necessary condition of equilibrium : 



j + fjL 2 $m 2 + etc. 



. ^ 0, (480) 



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

 are not, we may conceive of two entirely different substances meeting in equilibrium 

 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 absolutely unmixed, the amount of mixture can be increased, but is incapable of 

 diminution, and it is then consistent with equilibrium that the value of 5e (for a 

 variation 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 occurrence 

 of cases in which variations may be possible while the opposite variations are not. 



It will be observed that the sense in which the term reversible is here used is entirely 

 different from that in which it is frequently used in treatises on thermodynamics, 

 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. 



