60 EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 



moments. Now no change whatever in the state of the system, 

 which does not alter the value of the energy, and which commences 

 with the same, state in which the system was supposed at the com- 

 mencement of the short time considered, will cause an increase of 

 entropy. Hence, it will generally be possible by some slight variation 

 in the circumstances of the case to make all changes in the state 

 of the system like or nearly like that which is supposed actually to 

 occur, and not involving a change of energy, to involve a necessary 

 decrease of entropy, which would render any such change impossible. 

 This variation may be in the values of the variables which determine 

 the state of the system, or in the values of the constants which deter- 

 mine the nature of the system, or in the form of the functions which 

 express its laws, only there must be nothing in the system as modi- 

 fied which is thermodynamically impossible. For example, we might 

 suppose temperature or pressure to be varied, or the composition of 

 the different bodies in the system, or, if no small variations which 

 could be actually realized would produce the required result, we 

 might suppose the properties themselves of the substances to undergo 

 variation, subject to the general laws of matter. If, then, there is 

 any tendency toward change in the system as first supposed, it is a 

 tendency which can be entirely checked by an infinitesimal variation 

 in the circumstances of the case. As this supposition cannot be 

 allowed, we must believe that a system is always in equilibrium 

 when it has the greatest entropy consistent with its energy, or, in 

 other words, when it has the least energy consistent with its entropy. 



The same considerations will evidently apply to any case in which 

 a system is in such a state that AT; = for any possible infinitesimal 

 variation of the state for which Ae = 0, even if the entropy is not 

 the greatest of which the system is capable with the same energy. 

 (The term possible has here the meaning previously defined, and the 

 character A is used, as before, to denote that the equations are to be 

 construed strictly, i.e., without neglect of the infinitesimals of the 

 higher orders.) 



The only case in which the sufficiency of the condition of equit 

 librium which has been given remains to be proved is that in which 

 in our notation &/ = for all possible variations not affecting the 

 energy, but for some of these variations A^>0, that is, when the 

 entropy has in some respects the characteristics of a minimum. In 

 this case the considerations adduced in the last paragraph will not 

 apply without modification, as the change of state may be infinitely 

 slow at first, and it is only in the initial state that the condition 

 &7e = holds true. But the differential coefficients of all orders of 

 the quantities which determine the state of the system, taken with 

 respect of the time, must be functions of these same quantities. None 



