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the resultant of all these forces must be a vertical force, equal to the weight of the definite portion, and passing through its centre of gravity. For if these conditions are not satisfied, the definite portion of the liquid cannot be at rest; it would not be at rest even if it were solid, but would go up or down or turn round; and so it will not be at rest when it is liquid.

XXX. LIQUIDS STAND AT A LEVEL.

383. WE have shewn that the surface of a liquid in equilibrium in a vessel is a horizontal plane. Now suppose we put a liquid into a

vessel composed of two vertical tubes connected A by a horizontal tube. The surface of the liquid in each tube will be a horizontal plane as we have

B

already stated; and moreover the two surfaces will be in the same horizontal plane; thus if AB and CD denote the surfaces of the liquid in the tubes then AB and CD are in the same horizontal plane. This last fact we have not hitherto explicitly stated, though it is intimately connected with some of our previous results; the fact in its various forms is expressed by saying that liquids seek their level. It amounts to this: if liquid can pass from one vessel to another by means of a connecting channel it will do so, until the upper surface of the fluid is throughout in the same horizontal plane.

384. The preceding statement admits of easy experimental illustrations. It will be found, for instance, to hold with respect to a common tea-pot and its spout. If only a very small quantity of water is put into the teapot it may remain below the point of communication with the spout; but when more water is added it will pass into the spout, and then it will stand at the same level in the two parts of the vessel.

385. The fact is closely connected in theory with two others which have come before us. We have shewn in Art. 362 that the pressure at any point inside a liquid is in proportion to the depth of the point below the open surface; and we have shewn in Art. 372 that the pressure is equal at any two points in the same horizontal plane. Now these two statements would not be consistent with each other unless the liquid in communicating vessels stood at the same level. All the facts too are connected with the principle of Art. 184 that for stable equilibrium the centre of gravity should be as low as possible; for instance if the liquid in different communicating vessels did not stand at the same level, we could bring the centre of gravity of the whole to a lower position by taking liquid from the place where it stood highest and putting it into another vessel in a lower position.

386. So long as we keep within a few yards of the same spot on the earth liquid in a vessel or in a small pond has its surface practically a plane. But this is not true with regard to large expanses of water; we know for instance that the Pacific Ocean must be curved into a hemispherical form, and even for lakes of moderate size the deviation from a plane may be recognized. Thus, suppose a circular lake of four miles in diameter; if an accurately straight line could be made to pass from a point just in the circumference of the boundary to a point on the circumference diametrically opposite, it would dip under the surface of the water, and at the middle of the lake would be about 32 inches below the surface.

387. Thus for a large expanse of water the surface is not plane but curved. This leads us to give a strict definition of a level surface; it is such that at all points of it the force of gravity has the same value, and its direction is at right angles to the surface. The level surfaces are very nearly spherical in form round a common centre; the force of gravity is less at any point of the outer of two such surfaces than at any point of the inner.

388. The properties of liquids which we have considered produce various phenomena that are exhibited on the surface of the globe. Water confined in a pond

or lake maintains itself at rest, and takes a level surface, practically plane if the confined space is small, but otherwise curved into a nearly spherical form. On the other hand if there be an outlet for the water, as the particles have little cohesion they yield to the force of gravity and descend. Thus rain falling on the tops of mountains, if the soil is not soft and easily penetrable, collects in rills which unite and form larger streams. These descend along the sides of mountains and mix with others so as to produce rivers. The course is determined by the nature of the ground, and the general tendency of the water to descend. It is found that if the descent be about a foot in four miles the stream in a straight channel would flow at the rate of about four miles in an hour: the average slope of the large rivers of the world is greater than this. It belongs to Physical Geography to describe the various peculiarities which rivers present in their courses from the mountains in which they rise to the seas into which they fall, such as the cataracts which they form when they change their level suddenly and violently, and their occasional disappearance and reappearance after flowing for some time underground.

389. A canal is an artificial channel of water made to connect two places. If the two ends are not in the same level surface the entire course cannot be in one level surface; and even if the two ends are in the same level surface it may be difficult or impossible to construct the canal entirely on one level owing to the presence of mountains. Of course if the canal were one unbroken channel the water would descend from the higher parts leaving them dry, and would overflow the banks at the lower parts. To obviate this the canal in part of its course consists of separate portions called locks which stand at different levels, and which are separated from each other by flood-gates. When a boat is taken through this part of the canal a communication is opened between the compartment in which the boat is and that into which it is to pass the water in the two compartments is thus brought to the same level, the gates between them are opened and the boat is drawn onwards. Thus every time a boat passes up or down through the locks some water

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is lost from the highest part of the canal; and the supply must therefore be perpetually renewed by natural or artificial means.

390. The mode in which water is conveyed through our large towns offers an interesting exemplification of the principle that liquids stand at a level. A reservoir is formed at as high an elevation as the water is desired to reach; this is kept full by means of water falling into it or being pumped up into it from lower levels. Pipes proceed from the reservoir through the town which is to be supplied, and in any of these the water will rise to the height which it has in the reservoir; so that it can be brought to the upper rooms of tall houses. The ancient Romans were in the habit of bringing water to their towns from a distance, by means of aqueducts, that is by artificial channels constructed on a level surface, or on a gentle descent. Hence it has been supposed that they were not acquainted with the principle that liquids stand at a level; but it seems to be now made out that it was not ignorance of this principle but a want of the necessary pipes which kept them from using the modern system. Even in recent times the ancient system has been adopted as possessing some special advantages; an example is furnished by an aqueduct for supplying water to New York.

XXXI. VOLUMES OF SOLIDS IMMERSED IN LIQUIDS.

391. If we wish to determine the volume of a solid body of known regular form, as a cube or a sphere, we have only to use the rules for the process which are given in books on Mensuration. Thus, for instance, the solid body may be in the form of a brick 9 inches long, 3 inches broad, and 2 inches deep; and then we know that the volume is expressed in cubic inches by the product of the numbers 9, 3, and 2; that is the volume is 54 cubic inches. But rules cannot be given for finding the volume of any irregular body, as a stone or a coal. It is a natural consequence that we usually estimate the quantity of solids by weight rather than by volume, that is by pounds and ounces rather

than by cubic feet and inches. On the other hand liquids, by their property of yielding and filling all the corners of a vessel in which they may be placed, allow us to determine their volumes easily; and accordingly we usually estimate the quantity of liquids by volume.

392. But we may also use the fundamental property of liquids, namely their extreme mobility, to determine the volume of a solid. We suppose that the solid will sink in a certain liquid if left to itself; then if the solid be put into a vessel of the liquid it will displace liquid equal in bulk to its own. There are various forms in which this fact may be presented. Thus suppose a vessel of sufficient size just full of water; let a solid be carefully dropped in and the water which runs out accurately collected: then this water is obviously just equal in bulk to the solid. The volume of the water collected may be ascertained by pouring it into a vessel which has already been measured and has lines marked on its surface indicating how much it holds when filled up to an assigned level. Or again, take a vessel containing some water, though not full, and observe the level at which the water stands; then put in the solid, which we suppose to go to the bottom and to be perfectly covered by the water. The water now rises to a higher level than before, and the bulk of the solid is exactly equal to that of the water which would be comprised between the two levels. This quantity can be easily calculated if the vessel be of suitable shape; for instance, if the vessel have a rectangular base and its four sides vertical, the volume is found by the rule which we have already exemplified in Art. 391.

393. We have supposed the solid to sink in the water, but we know that many substances, as wood for example, will not sink in water. In this case we must press the solid into the water by a slender wire, or by other means. Or we may attach the solid to another of such a nature that both together will sink in water, and thus we can find the volume of both together; then we can find separately the volume of the sinker, and finally, by subtraction, the volume of the solid with which we are concerned.

394. In the same manner as we propose to find the whole volume of a solid we may also find the volume of any

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