1817.) Dilatation of Liquids at all. Temperatures. 435 
small variation may exist in this point in consequence of differences 
in the thermometers employed in the experiments, and likewise of 
the greater or lesser purity of the water examined ; for we have 
seen that the presence of foreign bodies in that liquid sinks the 
point of its greatest condensation, and even makes it entirely dis- 
appear; but our calculation, which approaches the mean of the 
experiments, leaves but a very small range to the true point. 
In a curious set of experiments made by Sir Charles Blagden to 
determine how far water in certain circumstances could be cooled 
down without freezing, he observed that its retrograde dilatation 
continued, and proceeded with such rapidity as to form a consider- 
able proportion of the total expansion which water undergoes when 
converted intoice. This is an evident consequence of our formulas. 
In the value of the apparent dilatation A,, when T is positive, a 
part of the terms destroy each other by the opposition of their 
signs ; but below 0°, T becoming negative, all the terms take the 
same sign, and must be added to each other. To know how far 
the difference can go, let us calculate the value of A; at + 10° R. 
and — 10° R. We obtain 
T = + 10° A, = 0:0001097 
T = — 10° A; = 0:0019188 
We see that the second is 18 times greater than the first. | 
Knowing the value of the true dilatation dry it is easy to deduce 
from it the apparent dilatation in vessels of any kind whatever ; for 
calling the cubic dilatation of the vessel K, the apparent dilatation 
A, is given generally by the equation 
> —KT 
Sl wih ake 
If we wish only to consider the dilatation A, for low tempera- 
tures, when it will always be small, we may neglect the product of 
3% — KT by K T, and suppose the denominator of the second 
member equal to unity. Then we have simply 
A,=%—KT '' 
We shall use it in this state for the purposes to which we mean to 
apply it. For greater simplicity, we shall substitute the letters a, 
b, c, tor the numerical coefficients contained in 2,3 that is to say, 
we shall take in general 
= aT+lT? + cT 
a, l, and c, having the values which we have just determined. 
Substituting this expression in A,, it becomes 
A, = (a — K)T + bT 4+ ¢T 
The apparent dilatation A, may be susceptible of a minimum, 
and the temperature at which it happens will depend upon the dila- 
tability of the vessel. The equation which determines this mini- 
mum is ; 
2EHQ 
