uncrystallized solid expands it does so in such a manner that its figure at one temperature is similar to that at another. Universal experience demonstrates the truth of this statement; and it can be very easily shewn that assuming it to be correct the cubical dilatation of a substance will then be as nearly as possible three times as great as its linear dilatation.

For let a represent the coefficient of linear expansion, and a' that of cubical expansion of the same substance for a rise of 1° Fahr. above 32°; also let L and V represent the length and volume of the substance at 32°. Then L (1 + a) and V (a) are its length and volume at 33°. But since similarity of figure is preserved, we shall have by a well-known proposition in geometry,

V : V (1 + a') : : L3 : Ľ3 (1 + a)3 ;

I : I + a' : : 1 : (1 + a)3 ;

I + a' = (1 + a) 3. = 1 + 3 a + 3 a2 + a3.

Now since a is a very small fraction we may dispense with the last two terms of the right hand member of this equation, and hence I + a' I + 3 a, or a' = 3 a nearly; that is to say,

=

the cubical is equal to three times the linear dilatation. 42. Increase of the coefficient of expansion with the temperature. It will be seen by comparing the mean coefficient of expansion between o° and 100°C with that between o° and 300°C, that the latter is greater than the former for each of the substances given in the above table; it would appear, however, that in the case of hardened steel the coefficient of expansion diminishes as the temperature increases; but this is probably due to the fact that heat deprives the steel of part of its temper, and that it thus becomes more like soft steel, which has a smaller coefficient of expansion than hard steel, as may be seen from the table of linear expansion already given.

DILATATION OF CRYSTALS.

43. It is found that many crystals do not expand under heat equally in all directions so as to preserve their similarity of figure. Mitscherlich has investigated at great length the action of heat upon crystals, and has obtained the following

1. Crystals of the regular system which do not cause double refraction dilate uniformly in all directions in the same manner as uncrystallized bodies.

2. Crystals that are optically uniaxal are differently affected. by heat in the direction of the principal axis and in the direction of the three secondaries, but in the direction of the latter they are similarly affected.

3. Crystals that are optically biaxal dilate unequally in all directions.

Mitscherlich believes he has determined, as the result of his investigations, that the tendency of heat in crystals is to increase the mutual distance of the molecules in that direction in which this is least, so as to equalize the distances in different directions and bring the axes into a state of equality.

REMARKS ON THE DILATATION OF SOLIDS.

44. The general law connected with the dilatation of solids is that enunciated at the commencement of this chapter, which states that such bodies expand when heated, but regain their original volume when they are restored to their original temperature.

Neither of these statements is, however, universally true, and a singular exception to the first occurs in the case of Rose's fusible metal. Erman, and afterwards Kopp, have found that there is for this body in the solid state a point of maximum expansion through heat, after which, if the

temperature be increased, it contracts instead of expanding. According to Kopp something of the same kind takes place in sulphur.

45. In the second place, the statement regarding the recovery by a solid of its original volume when it resumes its original temperature is by no means absolutely correct. For if a solid be cooled very suddenly, in most cases its particles have not had time to bring themselves into the condition proper to the reduced temperature, and in consequence the substance is in a state of constraint, which continues often for a very long time. This is probably the cause of the change of zero in a mercurial thermometer (Art. 22). For when such an instrument is made, or filled, the bulb is heated and suddenly cooled, and hence its particles have not had time to approach so near to one another as they would have done had the process of cooling been very gradual. The bulb is therefore abnormally dilated, and only recovers from this state after a considerable time, during which a slow contraction takes place and the mercury is pushed up in the tube, or the zero appears to rise.

In like manner when such an instrument is exposed to the temperature of boiling water and suddenly cooled, the bulb remains somewhat dilated, or the zero appears to have fallen, and only recovers its former position after ten days or a fortnight have elapsed.

Magnus, and afterwards Phipson, have noticed a similar behaviour in certain specimens of the idocrase and garnet family. These have their density considerably diminished after they have been heated to a red heat, but in the course of time they recover their former volume.

Other instances of this behaviour might be mentioned, and the knowledge of the fact is of much importance in many of the arts. It is accordingly well known to workers in the metals and in glass that the utensils which they form from

the molten material require to be very carefully and slowly cooled in order that the particles may have had time to assume their most stable position, otherwise the structure is fragile and comparatively useless. The process by which this is accomplished is called annealing.

It thus appears that time is an important element in the cooling of bodies; and with this reservation it may not perhaps be erroneous to assert that a solid body heated and very slowly cooled will regain its original volume on regaining its original temperature.

CHAPTER III.

Dilatation of Liquids.

The

46. Apparent dilatation and real dilatation. cubical dilatation of a liquid may be either apparent or real. By apparent dilatation is meant the apparent increase of volume of a liquid confined in a vessel which expands but in a less degree than the liquid which it contains. By real or absolute dilatation is meant the true change of volume of the liquid without reference to the containing vessel.

In order to find the real dilatation of liquids one of the following processes is employed.

47: (I) Method by thermometers. In this method the liquid under experiment is made to fill the bulb of a thermometer of which the internal volume or capacity is supposed to be known at the various temperatures of observation. This bulb is attached to a graduated stem, and the internal capacity of each division of this stem is likewise supposed to be known.

When this instrument has been filled with the liquid under examination it is exposed to different temperatures, and for each of these the position which the extremity of the liquid occupies in the stem is accurately noted. It is clear that by this means the volume of the liquid for each temperature becomes known, and hence the amount of its dilatation may be easily deduced.

48. (II) Method by specific gravity bottle. Here a vessel, the internal volume of which is accurately known for all temperatures, is separately filled at each temperature with the liquid under examination, and the whole is then weighed. The weight of the vessel when empty is also ascertained, and thus the weight of liquid which it contains at each temperature becomes known. But the volume of this liquid is also known; hence its density, or the weight of unity of volume, becomes known, and thus the dilatation may be determined. In this method the kind of bottle generally used is one made of glass, having a glass stopper which fits it accurately. This stopper is ground out of a capillary tube, such as that used for the stem of a thermometer, and hence, when the bottle is filled with liquid and the stopper pushed home, any excess of liquid is forced out through the capillary orifice. The bottle ought to be filled in this manner at a temperature lower than that of observation, so that, when it is subjected to the higher temperature and the liquid expands, the excess may escape by the orifice of the stopper and yet leave the bottle quite full.

49. (III) Method by weighing a solid in the liquid, or the areometric method. In this method a solid whose volume is accurately known for each temperature of observation is weighed immersed in the liquid at these temperatures. The difference between the weight of this solid in vacuo and its weight in the liquid will give us the means of determining the relative density of the latter at the