face parallel to the horizon; let it be depressed very gently until the weight of the displaced water is equal to the weight of the solid, and then be left to itself. The solid most probably will not remain in equilibrium, but will

turn over.

425. In order that a solid may be in equilibrium when floating on a liquid two conditions must be satisfied. (1) The weight of the solid must be equal to the weight of the liquid displaced. (2) The centre of gravity of the solid and the centre of gravity of the liquid displaced must be in the same vertical straight line. The first of these two conditions has been already explained. If a solid be wholly or partially immersed in a liquid it is acted on by two forces, its own weight vertically downwards, which may be supposed to act at its centre of gravity, and a force equal to the weight of the displaced liquid vertically upwards, which may be supposed to act at the centre of gravity of the displaced liquid. If these two forces are not equal the solid will move downwards or upwards according as the former or the latter force preponderates. But suppose that the two centres of gravity are not in the same vertical straight line, then even if the two forces are equal they do not keep the solid in equilibrium because they are not directly opposed to each other; they will turn the body round.

426. If a solid composed of materials lighter than water, bulk for bulk, is put into still water we know as an experimental fact that it will at last come to a position of equilibrium. There may be for a time a movement up and down, or a rocking to and fro; but the friction at last stops the motion, and the solid remains at rest. Also even if a body is composed of material which is heavier than water, bulk for bulk, yet by giving to it a hollow form we can in general secure for it a position of equilibrium when put on water. The subject is very important, and is connected with that of the stability and instability of equilibrium noticed in Art. 182.

427. If we suppose the floating solid to be symmetrical in shape, like a sphere, then it is easy to see that the centre of gravity of the floating solid and the centre of gravity of the water displaced do lie in the same vertical

straight line whatever may be the depth of immersion; and thus if this depth be suitably taken the solid will remain in equilibrium. The same remark applies to the brick-shaped solid when one face is kept horizontal. In such cases the equilibrium is stable so far as regards any movement up and down. For if the solid is pushed down a little the weight of the water displaced is greater than in the position of equilibrium; and so the upward force preponderates, and the solid rises when left to itself. In like manner if the solid be drawn up a little the weight displaced is less than in the position of equilibrium; and so the downward force predominates, and the solid descends when left to itself.

428. Let us suppose a ship or such like body floating in equilibrium on water. Let it be tilted by the wind or some other force sideways. Let ABC represent a vertical

G

M

section of the ship, taken at right angles to the length, passing through G, the centre of gravity of the ship, and cutting the keel at B. Let H be the centre of gravity of the water displaced by the ship in its tilted position. Then the ship is acted on by its own weight downwards at G, and by a force vertically upwards at H equal to the weight of the water displaced. If these forces are not equal there will be motion upwards or downwards; but this is of small consequence, because by such motion there is a tendency to promote the required adjustment for equilibrium, as explained at the end of the preceding Article. The important question is as to the direction in

T. P.

12

which the ship will turn round. Draw a vertical straight line through H, and let it meet BG produced, if necessary, at M. This point M is called the metacentre, and in books which discuss the theory of the subject it is shewn how the position of this point may be determined when the amount of tilting is very slight; but the process is not sufficiently elementary for our purpose. We may however easily see the importance which attaches to the position of the point M. Suppose M to be, as in the diagram, above G. Then it may be taken as tolerably obvious that the joint effect of the upward force at M and the downward force at G is to turn the ship back again so as to bring BG to be vertical as at first. Thus the original position of the ship is one of stable equilibrium with respect to this tilting. Suppose however that M falls below &; then in the same way we see that the joint effect of the upward force at M and the downward force at G is to turn the ship further away from the position in which BG is vertical. Thus the original position of the ship is one of unstable equilibrium with respect to this tilting. See Art. 345.

429. Hence we see that it is essential for the safety of a ship that the centre of gravity should not be too high up. The proper situation is secured by putting the heavy goods which the ship carries as low in the hold as possible. After a ship has discharged the cargo it is found necessary to put into the hold sand or stones or such things for the sake of bringing down the centre of gravity of the whole as low as possible; these things are called ballast. So also if people go on the water in a small boat they must be careful to remain sitting down so as to keep the centre of gravity low; and especially they should avoid any sudden rising, which may elevate the centre of gravity, and tilt the boat at the same time,

430. We may observe that we have not taken the most general form of the investigation. We have assumed that the body is of the nature of a ship so as to have its two sides symmetrical, and we have supposed that the tilting is from side to side. Under these circumstances G and H remain always in the same vertical plane in which the tilting takes place; otherwise the matter becomes too

complicated for an elementary book. As a simple example let us suppose a sphere of wood floating on water. The centre of gravity of the solid is the centre of the sphere; and it is a result of geometry that the metacentre is also at the centre of the sphere. Thus the equilibrium is of the kind which we have called neutral in Art. 183. If instead of a whole sphere the floating body is a portion of a sphere cut off by a plane, then whether this portion is greater or less than a hemisphere, the centre of gravity will be below the centre of the sphere, while the metacentre is at the centre of the sphere; hence the body will float in stable equilibrium when the flat part is horizontal and outside the water.

XXXVI. SPECIFIC GRAVITY OF SOLIDS.

431. We have often in the preceding Chapters spoken of one body as heavier than another, bulk for bulk; thus gold is more than nineteen times as heavy as water bulk for bulk. In other words a cubic inch of gold is more than nineteen times as heavy as a cubic inch of water; and so for a cubic foot. When we speak of one body as heavier than another we may mean heavier bulk for bulk; in this sense gold is heavier than iron. Or we may mean that one assigned body is heavier than another, as that a certain iron bar is heavier than a certain gold coin. It is always plain from the circumstances in which of these senses we use the word heavier; the former is usually the sense required in the present work. We sometimes use the words heavy and light as if there were no comparison intended between the body with which we are concerned and other bodies. Thus we may say that lead is heavy and that cork is light. But some comparison is really intended; we mean that lead is heavier, bulk for bulk, than most objects with which we are familiar; and that cork is lighter, bulk for bulk, than most objects, or at least than most kinds of wood, with which we are familiar.

432. We have already in Art. 403 defined specific gravity as the proportion of the weight of any substance to the weight of an equal volume of the standard substance;

and we have stated that the standard substance is usually water. But we must now be a little more precise with respect to this standard substance. Water as obtained from springs or rivers is not always the same thing; it contains various substances mixed with it in greater or less degree, and hence the condition is added that the water must be pure. Water is made pure by distillation, that is, the water must be boiled and the vapour collected and condensed by cooling: in this way it is found that the substances which common water holds in solution are left behind, and pure water obtained. Moreover the bulk of water changes as the temperature changes, other things being the same. It is found that pure water diminishes in bulk as the temperature diminishes until the temperature is about 40 degrees of Fahrenheit's thermometer, and after that if the temperature is still lowered the bulk increases. Hence the temperature of 40 degrees of Fahrenheit's thermometer is that which it is found convenient to take for the standard. Thus finally we may say that the specific gravity of any substance is the proportion of the weight of the substance to the weight of an equal volume of pure water at the temperature of 40 degrees of Fahrenheit's thermometer.

433. The words dense and density are often used in books on Natural Philosophy, and we may here exemplify the meaning of them. We say that water has its greatest density at the temperature of 40 degrees of Fahrenheit's thermometer, or that water is more dense at this temperature than at any other. The simple fact which we have to express is that a cubic foot of water at this temperature weighs more than a cubic foot of water at any other temperature. As a convenient mode of representing this to our imagination we may suppose that the particles of water are closer together at the standard temperature than at any other. The density of a given body then is greater the smaller the volume of that body is; thus if a body is brought by cold or by pressure to occupy half its original space we say that the density is doubled. It would not be easy to double the density of a solid or of a liquid; but the density of a gaseous body can be easily doubled or even still more increased. We sometimes extend the range of the words dense and density, and use them in the com