ae 
Se aD eae 
J. W. Gibbs—Equilibrium of Heterogeneous Substances. 445 
If, then, ¢ is known as a function of 4, p, m 
we can find 7, v, #,, fo, - + + My in terms of the | tite ‘uration 
By eliminating r we may obtain again n + 8 0 tage rela- 
tions between the same 2n +5 variables as at 
If we integrate (5), (6) and (8), supposes the quantity of 
the compound substance considered to vary from zero to any 
ne value, its nature and state remaining unchanged, we 
tai 
é=tyn—pv+ um, + Mm, ... MM, (9) 
po=-—prvt+um+um,... ot (10) 
c= wm, + wm, ... + f,m (11) 
If we differentiate (9) in the a general manner, and com- 
pare the result with (5), we obtai 
-vdp+ndt+m,du, MS oot im, OS 0, (12) 
or 
dp = dt dy, + "du, ane + dy,=0. (13) 
Hence, there is a relation between the n + 2 seta = t, P, 
Pay fay» - - fw Which, if known, will enable us to 
terms of these quantities a all the ratios of the n + 2 quasi 
av With (9), this will make n+ 38 inde- 
fenton! Siar between the same 2n + 5 variables as at first. 
ny equation, therefore, between the quantities 
&, 1; v, wey My + +» May 
= y, t, Vv; m,, Ma, seals! Mny 
or ¢; Zt, Ps mM, Migs ss Why 
nis t, Ps By, By +2 > Muy 
is a poracagees equation, and any such is kote; equivalent 
to any o 
— hases.—In considering the different Leaaoegians 
bodies which can be formed out of any set of co 
Stances, it is convenient to have a term which shall refer solely 
to the composition and thermodynamic state of any such body 
Without regard to its size or form. The word phase has been 
chosen for this purpose. Such bodies as differ in composition 
or state are called different phases of the matter considered, all 
sini The properties of the quantities —y and —¢ regarded as functions of the 
ed 
in a memoir entitled Sur les factions caractéristiques des divers fiuic r 
la théorie des vapeurs Apes Savants Etrang.,t xxii.) A brief sketch of his 
method in a form slightly different voy that uutmately adopted i is given in Comptes 
Rendus, t. xix, (1 oem) pe Ean 7, and by M. Bertrand 
in Comptes Ren —- t. kxxi, p. 257. of Massiou appears to have been the first to 
Solve the shang? of representing all th Ai hes of a body ‘of invariable com- 
