IN STUDYING QUESTIONS OF CHEMICAL MECHANICS. 359 
portion of acid « is always a fraction less than unity, which ren- 
ders the product (1 —«)< positive; 2, — 1 is also positive, since 
the density of the disaggregated acid surpasses that of the water. 
The relation 
(1 — «)e (8, —1) 
e+(1—e«)38, 
is therefore essentially positive by these conditions. But, at the 
temperature of 6°°8, which we are considering, the values of a! 
and of 6 are such, that we have 
100 + @ = — 38°0875, 100¢-+ b= — 302°703 + 100«. 
These two factors are therefore both negative, and they remain 
so at all observable temperatures, which renders their ratio posi- 
tive., Hence, under all the circumstances which the experiment 
may present, the difference %, — @,, is always positive; that is 
to say, that, generally, the real density %,, observed or calculated 
by the hyperbolic law, exceeds the density 3,, calculated hypo- 
thetically by the law of simple mixture. These densities occa- 
sionally become equal only in the two extreme cases in which 
we should suppose «= 1 or «=O. In fact, when «= 1, the 
system is entirely formed of disaggregated acid whose density is 
@,, a circumstance which we have taken as the common starting- 
point in the general expressions of %, and of 6,,. The other 
case, in which we take ¢ = 0, supposes that the solution under 
consideration contains no acid, or only an infinitely small quan- 
tity in comparison to the quantity of distilled water associated 
withit. Then 8, and 3,, should still be equal, as becoming both 
the one and the other equal to unit; and it is also to this com- 
mon value that their individual expressions are reduced, as may 
easily, be verified. 
47. Between these two extremes the difference 3, — 3,, attains 
a maximum of value corresponding to that of the variable factor 
(l—e)e : 
(l00e — 6) [e+ @—s)8]° 
in order to determine simply its conditions, I substitute for < a 
new variable n, so that we have 
(l—«) =ne, 
consequently 
