( 452 ) 
the substance is always assumed to fill the given volume homo- 
geneously, remains the same. But if we should doubt this, the 
assumption of the principle of continuity would enable us again to 
conclude to a similar form. 
This is the simplest way of uniting the two branches which may 
be experimentally realized, in the same way as for the isothermal 
the usual way of uniting the gas- and liquid branch is the simplest. 
And they are actually equivalent — one is the mathematical conse- 
quence of the other. 
So if Z and p are given, three values of ¢ belong to any mixture 
when p remains between the values of pj, and p,,, which values 
belong to that mixture, which is always assumed to be homoge- 
neous, at this value of 7. As soon therefore as 7 is above what 
might be considered as the critical temperature of such a mixture, 
these three values are reduced to one. In this that value of 7 is 
considered as critical temperature, for which the isothermal can show 
only one horizontal tangent for homogeneous phases. 
But for every mixture these three values of ¢ or that one value 
of £ depend on the composition and in general they will be different 
and that for two reasons 1st because of the fact that in the value 
of ¢ occurs the pure function of « and y, which indicates increase 
of entropy in the mixing; viz.: 
— MR §(t—«#—y) log (ley) + w log x + ylogy} 
and 224 because also the second part of ¢, viz.: 
Dr { pdv = pv — MRT log (vb) Hr Ze 
differs for the different mixtures, when they are all taken at the 
same value of p and 7, It is the value of this expression, which 
is represented as ordinate in fig. 1. When v can be calculated 
a Rite 
from v = — —, this ordinate is independent of the nature of the 
substance, and so has the same value for all mixtures in the rarefied 
gaseous state. The gas branches can therefore always coincide, at 
least as long as p is exceedingly small. But as soon as the degree 
of density is such that they are no longer perfect gases, these 
lines deviate — and in the liquid state the difference of the ordi- 
nates for two substances can become so great, that the difference 
mentioned sub 1 is quite insignificant compared to it. Therefore 
