114 '/. W. Gibhs — Equilibrium of Heterogeneous Substances. 



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

 higher orders.) 



The only case in which the sufficiency of the condition of equi- 

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

 in our notation dj] ^ for all possible variations not affecting the 

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

 entroj^y 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 

 Sr^^ -S 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 of these differential coefficients can have any value other than 

 0, for the state of the system for which 8ri^ ^0. For otherwise, as 

 it would generally be possible, as before, by some infinitely small 

 modification of the case, to render impossible any change like or nearly 

 like that which might be supposed to occur, this infinitely small 

 modification of the case would make a finite difference in the value 

 of the differential coefficients which had before the finite values, or 

 in some of lower orders, which is contrary to that continuity which 

 we have reason to expect. Such considerations seem to justify us 

 in regarding such a state as we are discussing as one of theoretical 

 equilibrium ; although as the equilibrium is evidently unstable, it 

 cannot be realized. 



We have still to prove that the condition enunciated is in every 

 case necessary for equilibrium. It is evidently so in all cases in 

 which the active tendencies of the system are so balanced that 

 changes of every kind, except those excluded in the statement of 

 the condition of equilibrium, can take place reversibly, (i. e., both in 

 the positive and the negative direction,) in states of the system dif- 

 fering infinitely little from the state in question. In this case, we 

 may omit the sign of inequality and write as the condition of such a 

 state of equilibrium 



(0»,rr:0, i.e., {6e\ = (10) 



But to prove that the condition previously enunciated is in every 

 case necessary, it must be shown that whenever an isolated system 

 remains without change, if there is any infinitesimal variation in its 

 state, not involving a finite change of position of any (even an infini- 

 tesimal part) of its matter, which would diminish its energy by a 



