THERMODYNAMICAL SYSTEM OF GIBBS 77 



infinitely slow, there would be no rea8on to disallow it because 

 it can be entirely checked by an infinitely small modification of 

 the case. The argument depends finally on the consideration 

 that an infinitely small modification of the circumstances cannot 

 cause a finite change in the rate of change of the system, for as 

 is explicitly stated in a succeeding paragraph, this is "contrary 

 to that continuity we have reason to expect." 



"The same considerations will evidently apply to any case in 

 which a system is in such a state that A17 ^ 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." Thus a system of hydrogen, oxygen and 

 water is in equilibrium when (Atj), ^ 0, for all possible varia- 

 tions, even if the entropy is not the greatest for the same amount 

 of energy. The conditions may be such that the combination 

 of hydrogen and oxygen to water would cause an increase of 

 entropy in the isolated system, but if this change is prevented 

 by passive forces or resistances to change, variations involving 

 it are not possible, and the system is in equilibrium if (At?)^ ^ 0, 

 for all variations which do not involve such changes. 



(c) When "677 ^ for all possible variations not affecting 

 the energy, but for some of these variations At? > 0, that is, 

 when the entropy has in some respects the characteristic 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 

 {dr])t ^ holds true." None of the differential coefficients of 

 all orders of the quantities which determine the state of the 

 system, taken with respect to the time, can have any value 

 other than 0, for the state of the system for which (5r?), ^ 0. 

 For if some of them had finite values, "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 modifica- 

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

 differential coefficients which had before the finite values, or 

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



