various temperatures. This will be seen from the following example:

Let us suppose that the volume of the solid at 32° Fahr. is denoted by unity, but at 212° Fahr. by 1.006. Suppose also that the apparent loss of weight of the solid when weighed in the liquid at 32° is 1800 grains, while at 212° the same is only 1750 grains: 1800 grains is therefore the weight at 32° of a volume of the liquid equal to unity, while 1750 grains is the weight at 212° of a volume of the liquid equal to 1.006. Hence 1739.56 grains will denote the weight of unity of volume of the liquid at 212°; and hence also 1800 grains, which at 32° occupied a volume equal to unity, will

at 212° occupy a volume

1800 1739.56

=

1.0347; or the dilata

tion between these two temperatures is represented by .0347. 50. Absolute dilatation of mercury. In all these methods the capacity of the vessel or the volume of the solid employed must be known at the various temperatures of observation, or, in other words, we must know its cubical dilatation.

But the remarks in the preceding chapter (Art. 38) lead us to conclude that in order to determine accurately the cubical dilatation of a solid it is hardly sufficient to determine the linear dilatation of another specimen of the same material and to multiply this by three, but the cubical dilatation ought, if possible, to be obtained by direct experiment. We have already seen (Art. 39) that in order to accomplish this it is necessary to know the absolute dilatation of some one fluid, such as water or mercury.

The problem before us is thus reduced to the determination of the absolute expansion of some one liquid, after which that of other liquids may be easily derived. This therefore is a determination of much importance; and since mercury has been chosen for the purpose, we shall now

proceed to shew how the absolute expansion of this liquid may be found.

51. The method about to be described was first employed by MM. Dulong and Petit. It consists in filling a U-shaped tube with mercury, one limb being kept at a low and the other at a high temperature. The portion of the liquid which is heated will of course be specifically lighter than the other, and hence the hot column must be higher than the cold one, since the two balance each other hydrostatically. Thus if D, D' are the two densities, and H, H' the corresponding heights, we shall have D : D' :: H': H, or the heights will vary inversely as the densities. This method is perfect in principle, but it is almost impossible to keep a column of mercury at a constant high temperature and at the same time be able to observe accurately the position of the top of the column. Regnault has however improved the apparatus so as to overcome this obstacle, and the following sketch will give an idea of the arrangement which he employed. ab, a b' are the two vertical tubes to be filled with mercury, and these are connected together near the top by a horizontal tube a a'. At the bottom they are not connected together, but ab is connected with the horizontal tube bc, and a' b' with b'c'. To the extremities of these horizontal tubes two vertical glass tubes cg, c'g' are attached, and these are both connected with a tube hi leading to a large reservoir f supposed to be filled with gas whose temperature is constant; hence the pressure of this gas in the tubes cg, c'g' is also constant. Heat is applied to the tube ab, and by means of an agitator every part of this tube, including the mercury which it contains, may be brought to the same temperature throughout, and the value of this temperature is accurately ascertained. On the other hand, the tube a'b' is exposed to a current of cold water of a known constant temperature.

The tubes a b, a'b' are supposed to be filled with mercury until above the level a a', but we will shew in the sequel that it is not necessary to know the height of the fluid above this level.

Now let p denote the whole pressure due to the left hand column of mercury, and p' the whole pressure due to the right hand column. The pressure at c is evidently p, while that at c' is p'. Hence the pressure at d = p-pressure of column cd, and in like manner pressure at d'p'- pressure of column c'd'. But the pressure at d is equal to that

at d', both being equal to the pressure of the gas in the reservoir ƒ: hence we have.

p-pressure of c d = p' — pressure of c' d',

and therefore

=

(1)

p-p pressure of column (c' d' — c d) .. that is to say, the difference between the pressures of the two great vertical columns is equal to the pressure of the column of mercury contained between the levels d'and d. Now since the tubes ab, a'b' communicate together by a a', it is evident from hydrostatical principles that the portions of the two vertical columns above a a' are in equilibrium with each other, and therefore that the pressures of these two portions are equal. But p, or the whole pressure of the left hand column, = pressure of column ab + pressure of portion above a; and in like manner p' = pressure of column ab'+ pressure of portion above a. Now since the pressures above a a' are equal, it follows that

p-p= pressure of a b' — pressure of a b,...

and equating (1) with (2),

pressure of (c'd' - cd) = pressure of a'b'

(2)

pressure of a b. We have thus obtained an expression for the difference in pressure between two columns of mercury (a b, a' b') of equal length but of different temperatures, and since there is no occasion to view the top of the column, we can perfect our arrangements for keeping the whole at the same temperature throughout, while by the insertion of an air thermometer alongside of a b this temperature may be measured with great exactness. By this means therefore the relative density of mercury at various temperatures may be determined, and its dilatation thence easily deduced.

52. Using this method, and also a modification of it, Regnault obtained results which enabled him to construct a table giving the dilatation of mercury for every ten degrees Centigrade from o°C to 350°. But before exhibiting this

table let us explain the distinction between the mean and the true coefficient of dilatation, as it is quite necessary to know this in the case of liquids which change their rate of expansion from one temperature to another.

In general language, if we take a quantity of liquid whose volume at o°C is equal to unity, then the true coefficient of dilatation of this liquid at any point is the rate of increase in volume of the liquid at that point, as the temperature goes on regularly increasing.

On the other hand, the mean coefficient of dilatation for 1°C of the liquid between o° and any point is the mean rate of increase in volume of the liquid between these two points, that is to say, it is the whole expansion divided by the number of degrees included between the two points.

Thus we see in the following table, 2nd column, that the whole dilatation of mercury between o° and 100°C is 018153; that is to say, a volume of this fluid equal to unity at o° will at 100° be equal to 1.018153. Now .018153 is the increase for 100°, and hence the mean increase for 1° will be the hundredth part of this or .00018153, which accordingly will be found in the third column opposite 100°, as denoting the mean coefficient of dilatation of mercury between o° and that point.

On the other hand, the true coefficient of dilatation of mercury at 100° is found by the fourth column to be .00018405; that is to say, if the temperature rises through a very small distance such as 1° and becomes 101°C there will be an increase of volume represented by .00018405.