We shall think the vapour phase denoted by 1, the two liquid 
phases by 2 and 3; now w, and x, become equal in the critical 
end-point, but also 1, and 5, and v, and v,; so both the numerator 
and the denominator become zero. It is in this that this case distinguishes 
itself from the three-phase equilibrium solid-liquid-vapour, as VAN DER 
Waats observes, loc. cit.') because when the concentration of solid 
phase and liquid become equal, their volume and their entropy are 
ae ‚dp. 
not equal. We arrive at a determination of ae in the following way. 
The critical end-point is a plaitpoint for which the phases 2 and 3 
have coincided. Now we can represent the difference of the volume 
and the entropy between coexisting phases in the neighbourhood of 
a plaitpoint by : 
| dv \ dv 
Vv, = Vz T (w, —#,) An Toe and Yo — Ws as (eZ) | a +... 
vs pT On, pr 
So we may put for the limit: 
ie we 
Be Oy neren, ek ne rn 
a, p uw 2 jdt 
; : Jd, a 
If we substitute these values in the equation of ak and if we 
¢ 323 
divide numerator and denominator by wv, —.#,, we get: 
( 07 
Ns — (Ut > 
dp ed hi 13 1 3) dn, PT We. a) 
On the other hand for the plaitpoint which arises by the coinei- 
dence of «, and x,, the following equation holds: 
dT 
Oy 
nh — («,—x,) | —— 
pl. 23 29 B : 
or if we apply again the same expansion into series as above, but 
take the terms with (v,—.,)? into consideration : 
On? 
dp 7 
En — - . . . . . . (2) 
a Ge ) 
1) Van DER WAALS. These Proc. X. p. 191. 
