444 J. W. Gibbs—EKquilibrium of Heterogeneous Substances. 
y May ee ly 
pendently variable components, and p,, fa, - he poten- 
tials for these components. It is easily shown that ¢ is a 
1) M,,.  . m,, and that the complete value 
of de is given by the equation 
de=tdn—pdv + y,dm,+ yu,dm,... + p,dm,. (5) 
Now if ¢ is known in terms of 7, v, m -. mM, We can 
obtain by differentiation ¢, p, w#,, ... #, in terms of the same 
variables. This will make n + 8 independent known relations 
between the 2n +5 variables, ¢, 4, v,m,,™m,,.-- My 4 P; 
ese are all that exist, for of these varia- 
require consideration. A single equation from which all these 
relations may be deduced may be called a fundamental equa- 
tion. An equation between ¢, 7,v,m,,m,, .. - mM, isa funda- 
mental equation. But there are other equations which possess 
the same property. 
If we suppose the quantity ¢ to be determined for such a 
lass as we are considering by equation (3), we may obtain by 
differentiation and comparison with (5) 
dp =—ndt—pdv + w,dm,+m,dm,... + 1,dm,. (6) 
If, then, ¢ is known as a function of ¢, v, m,, Mg, - + + Mn We 
can Hnd %, p, ft, #,,... pw, in terms of the same variables. 
Tf we then substitute for ¢ in our original equation its value 
taken from equation (3) we shall have again n + 8 independent 
Pe between the same 2n + 5 variables as before. 
et 
Cet v (7) 
then, by (5), Ue ed 
= —ndt+vdp+p,dm,+ y,dm,...+Hadm, (8) 
ea a 
Ben alee 
Sener wen 
