POSITIONS OF THE TWO CENTRES.

109

becomes exactly equal to the weight of an equal volume of the liquid, there will be equilibrium; but, unlike the equilibrium in the experiment in § 75, this will evidently be unstable, for a slight movement either upwards or downwards will alter the resultant force so as to produce further movement in the same direction.

77. Relative Positions of the Centre of Gravity and Centre of Buoyancy. -In order that a floating body, wholly or partially immersed in a liquid, may be in equilibrium, it is evidently necessary that its weight be equal to the weight of the liquid displaced.

This condition, which is absolutely necessary, is, however, not sufficient; we require, in addition, that the action of the upward pressure should be exactly opposite to that of the weight; that is, that the centre of gravity and the centre of buoyancy be in the same vertical line; for if this were not the case, the two contrary forces would compose a couple, the effect of which would evidently be to cause the body to turn.

In the case of a body completely immersed, it is further necessary for stable equilibrium that the centre of gravity should be below the centre of buoyancy; in fact we see, by Fig. 69, that in any other

Relative Positions of Centre of Gravity and Centre of Pressure.

position than that of equilibrium, the effect of the two forces applied at the two points G and O would be to turn the body, so as to bring the centre of gravity lower. But this is not the case when the body is only partially immersed, as most frequently happens. In this case it may indeed happen that, with stable equilibrium, the centre of gravity is below the centre of pressure; but this is not necessary, and in the majority of instances is not the case. Let Fig. 70 represent the lower part of a floating body—a boat, for instance. The centre of pressure is at O, the centre of gravity at G, considerably above; if the body is displaced, and takes the position shown in the figure, it will be seen that the effect of the two forces acting at O and at G is to restore the body to its former position. This difference from

upon

what takes place when the body is completely immersed, depends the fact that, in the case of the floating body, the figure of the liquid displaced changes with the motions of the body, and the centre of buoyancy moves towards the side on which the body is more deeply immersed. It will depend upon the form of the body whether this lateral movement of the centre of buoyancy is sufficient to carry it beyond the vertical through the centre of gravity. The two equal forces which act on the body will evidently turn it to or from the original position of equilibrium, according as the new centre of buoyancy lies beyond or falls short of this vertical.1

78. Advantage of Lowering the Centre of Gravity. Although stable equilibrium may subsist with the centre of gravity above the centre of buoyancy, yet for a body of given form the stability is always increased by lowering the centre of gravity; as we thus lengthen the arm of the couple which tends to right the body when displaced. It is on this principle that the use of ballast depends. 79. Phenomena in apparent Contradiction to the Principle of Archi

medes.-A body cannot float in a liquid unless it have a density less than that of the liquid. This natural consequence of the principle of Archimedes seems at first sight to be contradicted by some wellknown facts. Thus, for instance, if small needles are placed carefully on the surface of water, they will remain there in equilibrium (Fig. 71). It is on a similar principle that several insects walk on water (Fig. 72), that a great number of bodies of various natures, provided they be very minute, can, if we may so say, be placed on the surface of a liquid without penetrating into its interior. These curious facts depend on the circumstance that the small bodies in

1 If a vertical through the new centre of buoyancy be drawn upwards to meet that line in the body which in the position of equilibrium was a vertical through the centre of gravity, the point of intersection is called the metacentre. Evidently when the forces tend to restore the body to the position of equilibrium, the metacentre is above the centre of gravity; when they tend to increase the displacement, it is below. In ships the distance between these two points is usually nearly the same for all amounts of heeling, and this distance is a measure of the stability of the ship.

We have defined the metacentre as the intersection of two lines. When these lines lie

LIQUIDS IN SUPERPOSITION.

111

question are not wetted by the liquid, and hence, in virtue of principles which will be explained in connection with capillarity (Chap.

xi.), depressions are formed around

them on the liquid surface, as represented in Fig. 73. The curvature of the liquid surface in the neighbourhood of the body is very distinctly shown by observing the shadow cast by the floating body,

when it is illumined by the sun; it is seen to be bordered by luminous bands, which are owing to the refraction of the rays of light in the portion of the liquid bounded by a curvilinear surface.

M

D

B

The existence of the depression about the floating body enables us to bring the condition of equilibrium in this special case under the general enunciation of the principle of Archimedes. Let M be a section of the body, CD the distance to which the depression extends, and AB the corresponding portion of any horizontal layer; since the pressure at each of the points of AB must be the same as in the other parts of the layer, the liquid acts in exactly the same way as if M did not exist, and the cavity were filled by the liquid itself.

Fig. 73.

We may thus say in this case also that the weight of the floating body is equal to the weight of the liquid displaced, understanding by these words the liquid which would occupy the whole extent of the depression due to the presence of the body.

80. Liquids in Superposition. When liquids of different densities, which do not readily mix, are placed in the same vessel, the particles of the denser liquids unite and fall to the bottom, just as a solid body sinks in a liquid of less density; finally, the liquids arrange themselves in the order of their respective densities, the surfaces of separation being horizontal. This fact is verified by means of the phial called the phial of the four elements. It is a flask (Fig. 74) containing mercury, water, and oil. In the state of equilibrium the mercury is at the bottom, the oil at the top, and the water in the middle; if the in different planes, and do not intersect each other, there is no metacentre. This indeed is the case for most of the displacements to which a floating body of irregular shape can be subjected. There are in general only two directions of heeling to which metacentres correspond, and these two directions are at right angles to each other. For an investigation of the conditions of stability in floating bodies, see Thomson and Tait's Natural Philosophy, §§ 763-768.

flask is shaken, the liquids are for the moment mixed, but in returning to repose do not fail to resume their former positions.

It is easily seen from the ordinary rules of hydrostatics, that the surface of separation of two different liquids must be horizontal. Let there be two liquids in a vessel (Fig. 75); the free surface is necessarily horizontal. If now we take two equal superficial elements n and n' in a horizontal layer of the lower liquid, they must be subjected to equal pressures; these pressures are measured by the weights of the liquid cylinders nrs n'tl; and these latter cannot be equal unless there be the same height of the lower liquid above the elements n and n'. This reasoning holds for all points in the horizontal layer, which must therefore be at a constant distance from the surface of separation; in other words, this surface must be horizontal.

Fig. 74.

Phial of the Four Elements.

This property is liable to considerable modification in the case of

Fig. 75.

liquids which can dissolve each other or act chemically upon each other. Thus, if alcohol be carefully poured upon water in a glass, the two liquids will be seen to have for their surface of junction a horizontal plane; but on agitation a single liquid will be formed by their mutual action, and the separation will not again take place.

If the agitation is not sufficiently great, this intimate mixture will only partially ensue, and will be confined to the neighbourhood of the surface of contact. Two uniform layers of liquid will thus be formed, separated by an intermediate zone of unequal density. This is the case at the mouth of a river, where the fresh water forms on the surface of the sea a layer, the base of which is a compound of fresh and salt water.

CHAPTER X.

APPLICATION OF THE PRINCIPLE OF ARCHIMEDES TO THE

DETERMINATION OF DENSITIES.-HYDROMETERS.

81. Determination of Densities. We have seen in Chap. vii. that in order to determine the density of a body it is only necessary to measure the ratio existing between the weight of a certain volume of the body and the weight of an equal volume of water. The principle of Archimedes enables us to effect this measurement very easily, and the process which it suggests is sometimes more convenient than that which has been described in the chapter mentioned above.

(1.) Solid bodies.-Suppose that the object whose density we wish to determine is a piece of copper. It is suspended by a very fine thread to one of the scales of a balance (Fig. 76), its weight is determined, and found to be, say 125 358. The body is then immersed in water; the equilibrium is destroyed on account of the upward pressure of the water, and in order to re-establish it, we must add a weight of 14.248 to the scale supporting the body. This additional weight, according to the principle of Archimedes, represents the weight of a volume of water equal to the volume of the body. 125.35 14.24

The density of copper is thus

=8.8.

(2.) Liquid bodies.—From one of the scales of the balance is suspended (Fig. 77) any body whatever, which must, however, not be capable of being attacked by the liquids in which it is to be immersed; a ball of glass weighted inside with mercury is very well adapted to this purpose. The exact weight of this is obtained; it is then immersed in the liquid whose density is sought-alcohol, for example; an upward pressure is thus produced, and in order to reestablish equilibrium, a weight of 35.438. must be added to the scale.