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38. Remarks on the preceding table. If we suppose that by means of the methods already described a great amount of accuracy of measurement may be obtained, yet there is an uncertainty regarding the real temperature of the experimental bar, and this becomes very great for temperatures above the boiling point of water. In such cases, where a bath is used, it is not only very difficult to keep this at a constant temperature, but it is also very difficult to estimate accurately the temperature by means of a thermometer. This uncertainty with regard to estimation applies still more strongly to higher temperatures. But for the range between freezing and boiling water, which is that of the above table, it may perhaps be assumed that the determinations are very good. Whence then proceed the differences between the results of different observers, and even between those of the same observer when estimating the dilatation of different specimens of the same substance? This is probably due to two causes. In the first place, substances which bear the same name are not always of precisely the same chemical composition. Of these, glass may be mentioned as a prominent example, and accordingly we find the dilatation of this substance ranging in the table from .000918 to .000776. Brass, cast iron, and

steel are likewise compounds of which the composition is variable. But besides this, the commercial varieties of those substances which when pure are elementary, such as iron, lead, silver, gold, &c., often contain a very appreciable amount of impurity, so that the composition of different specimens is by no means uniform. Very often too a comparatively small impurity causes a very great alteration in some of the properties of a metal. In the next place, it ought to be observed that two solids may have precisely the same chemical composition, while yet their molecular condition may be different, owing to a difference in the treatment which they have experienced. Thus steel heated and suddenly cooled is a very different substance from steel which has not been treated in this manner; and accordingly we find that while steel tempered yellow has for its expansion .001240, untempered steel has .001080. Glass also will behave in a different manner according as it is annealed or unannealed, and in certain cases it is almost impossible to obtain two bars, although made of precisely the same material, which shall in all their properties be precisely alike.

39. Cubical dilatation of solids. To determine the cubical dilatation of a solid we may either, first, weigh the substance at different temperatures in a liquid of which the absolute dilatation is known, or we may, secondly, enclose it in a glass vessel the remainder of which is filled with mercury or water; and if the absolute dilatation of either of these liquids is known, that of the glass envelope and of the enclosed solid may be easily determined. To illustrate this second method let us in the meantime regard the dilatation of mercury as known, and suppose that the following experiment has been made.

A glass bottle thoroughly cleansed, so as to admit of being well filled with mercury without air specks, is found to hold

at 32° Fahr. 10169.3 grains of this fluid, while at 212° it only holds 10011.4 grains. Now it is known from Regnault's experiments (see Art. 52) that the dilatation of mercury between 32° and 212° = .018153, that is to say, a quantity of this fluid occupying a volume equal to unity at 32° will at 212° occupy a volume = 1.018153. Hence the weight of mercury occupying a given volume at 212° will bear to that occupying the same volume at 32° the proportion of 1: 1.018153, and hence (had the bottle not dilated) the weight of mercury filling it at 212° would have been 10169.3 = 9987.9 grains. But the glass envelope having 1.018153 expanded, the bottle holds 10011.4 grains, or 23.5 grains more than it would have held had there been no expansion. The volume of the expanded bottle will therefore bear to that of the same bottle at 32° the ratio of 10011.4 to 9987.9, or of 1.00235 to 1; and hence the expansion of this bottle between 32° and 212° will be .00235.

Let us now suppose that this bottle contains a piece of iron weighing 2000 grains, and that the remainder of it is filled with 6707.8 grains of mercury at 32°, while at 212° the mercury filling it only weighs 6599.4 grains. There is thus the loss of 108.4 grains of mercury between the two temperatures. Had there been no expansion either of the bottle or of the iron the amount of mercury sufficient to fill the bottle 6707.8 at 212° would have been 1.018153

=

6588.2 grains, and

there would thus have been the loss of 119.6 grains of this fluid. But we have already seen that the expansion of the bottle enables it to contain 23.5 additional grains of mercury, and hence, had the bottle expanded but not the iron, the loss would only have been 119.6 23.5 96.1 grains. The difference between this and the actual loss (108.4 grains) must therefore have been caused by the expansion of the

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iron, and this must be such that a piece weighing 2000 grains will expand between 32° and 212°, so as to occupy an additional volume equal to that occupied by 12.3 grains of mercury at 212°. If we assume that the specific gravities of mercury and of iron at 212° are 13.2 and 7.8, we shall find that this additional volume is that occupied by 7.26 grains of iron. Hence of the whole 2000 grains of iron at 212°, 7.26 grains are occupying an additional volume, while the remaining 1992.74 occupy the same volume as that originally occupied by the 2000 grains at 32°. Hence also

=

7.26

1992.74

.00364 denotes, according to this experiment, the cubical dilatation of iron between 32° and 212°.

40. This example is sufficient to give an idea of the process employed, which may be used for such substances as do not act chemically upon mercury, and even for those metals which do so, provided that they are protected by a thin film of their oxide.

The following table exhibits the cubical dilatations obtained after this method by different observers; of these MM. Dulong and Petit, and also M. Regnault, employed mercury as their fluid, while, on the other hand, Kopp made use of a flask filled with water.

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Other substances besides those mentioned in the above table have engaged the attention of observers. M. Brunner (fils) has made experiments on the cubical dilatation of ice. His method consisted in determining the specific gravity of this substance at different temperatures. From these experiments he concludes that, while at o°C ice has a density of 0.91800, at - 19°C its density has increased to 0.92013. This would give a cubical dilatation for 1°C of .000122.

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41. Remarks on the above tables. Relation of cubical to linear expansion. In comparing this last table with the preceding one of linear expansion we obtain the following result.

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From this it will be seen that the cubical expansion is in every case equal to about three times the linear expansion of the same substance. The reason of this relationship between the two follows at once from the fact that when an

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