po 
St. ae ee 
J. W. Gibbs— Equilibrium of Heterogeneous Substances. 449 
respect to discontinuous changes. These limits are in general 
distinct, but touch each other at critical phases. 
Geometrical tllustrations,—In an earlier paper,* the author 
has described a method of representing the thermodynamic 
equation cannot be completely represented by any surface or 
finite number of surfaces. In the case of three components, if 
e—Em 1 ton ™: 1" 
¢ log — = H+a og |? (17) 
that in ¢, ¢, v, and m is 
ip = Bm mt(c—H-clogt+alog™): (18) 
that in p, ¢ and pis 
Hne-a cha p—R c 
p=ae Sapling = ; ets) 
where e denotes the base of the Naperian system of logarithms. 
As for the other constants, c denotes the specific heat of the 
* Transactions of the cademy, vol. ii, part 2. 
Am. Joon. Scr.—Tuirp> — Vou. XVI, No, 96.—Dec., 1878. 
