J. W. Gibhs — Equilihrmm of Heterogeneous l^ubstuHces. 171 



when we give these variables these constant values, as when we 

 adopt any other of the suppositions mentioned above in re<^ard to the 

 quantities remaining constant. And in all these relations we may- 

 interchange in any way //, >«,, . . . «?„, if we intercliange in the same 

 way t, p(^, . . . i.i„. It follows that, within the limits of stability, 

 when we choose for anj^ one of the differential coefficients 



dt dii J c///„ 



d7f dw^; ' ' ' dm„ (^^1) 



the quantities following the sign d in the numerators of the others 

 together with v as those which are to remain constant in diiferentia- 

 tion, the value of the differential coefficient as thus determined will 

 be at least as small as when one or more of the constants in differen- 

 tiation are taken from the denominators, one being still taken from 

 each fraction, and v as before being constant. 



Now we have seen that none of these differential coefficients, as 

 determined in any of these ways, can have a negative value within 

 the limit of stability, and that some of them must have the value zero 

 at that limit. Therefore, in virtue of the relations just established 

 one at least of these differential coefficients determined by considerino- 

 constant the quantities occurring in the numeratoi-s of the others 

 together with v, will have the value zero. But if one such has the 

 value zero, all such will in general have the same value. For if 



for example, has the value zero, we may change the density of the 

 component S„ without altering (if we disregard infinitesimals of 

 higher orders than the first) the temperature or the potentials, and 

 therefore, by (98), without altering the pressure. That is, we may 

 change the phase without altering any of the quantities t,j), /<j, , . . 

 /,/„, (In other words, the phases adjacent to the limits of stability 

 exhibit apj^roncimateli/ the relations characteristic of neutral equili- 

 brium.) Now this change of phase, which changes the density of 

 one of the components, will in general change the density of the 

 others and the density of entropy. Therefore, all the other differen- 

 tial coefficients formed after the analogy of (182), i, e., formed from 

 the fractions in (181) by taking as constants for each the quantities in 

 the numerators of the others together with u, will in general have 

 the value zero at the limit of stability. And the relation which 

 characterizes the limit of stability may be expressed, in general, by 

 setting any one of these differential coefficients equal to zero. Such 



