J, W. Gihhs — Equilihrkmi of Heterogeneous Substances. 113 



state of the system must take place, but these will necessarily be 

 such that the energy and the entropy will remain unchanged and 

 the system will continue to satisfy the same condition, as initially, of 

 having the greatest entropy consistent with its energy. Let us con- 

 sider the change which takes place in any time so short that the 

 change may be regarded as uniform in nature throughout that time. 

 This time must be so chosen that the change does not take place in it 

 infinitely slowly, which is always easy, as the change which we sup- 

 pose to take place cannot be infinitely slow except at particular 

 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 Avill generally be possible by some slight varia- 

 tion 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 teraperatiire or pressure to be varied, or the composition of 

 the diiFerent 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 A;? ^ for any possible infinites- 

 imal variation of the state for which Ae= 0, even if the entropy is 

 not the least 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 



Trans. Conn. Acad., Vol. III. 15 October, 1875. 



