376. Let ABCD represent a vertical side of a vessel, which is in the form of a rectangle; AB is supposed at the bottom, and CD at the surface of the liquid. Let

EF be parallel to the top and bottom, and midway between them. Take two equal and parallel strips of the side, one as much below EF as the other is above it; the pressure on the former is greater than it would be for an equal strip close to EF, and the pressure on the latter is just as much less. Hence the sum of the pressures on the two strips is the same as if they were both placed close to EF. Proceeding in this way we see that the pressure on points in EF may be called the average pressure all over the side; and the whole pressure is the same as if the whole side were at the depth of EF. Thus the whole pressure on the side is equal to the weight of a column of the liquid having the vertical side for base, and half the depth of the side for height. For instance, if the vessel is a cube open at the top and full of liquid, the whole pressure on one side is just half the pressure on the base.

377. The pressures on all parts of the plane side are parallel, being all at right angles to the plane; hence in this case the whole pressure is the same thing as the resultant pressure: see Art. 166.

378. We know that for every system of parallel forces there is a centre at which the resultant of the whole system may be supposed to act: see Art. 166. When the parallel forces are the pressures of a liquid on a plane this point is called the centre of pressure. In the case in which the plane is the rectangular side of a vessel full of liquid the position of the centre of pressure can be easily determined.

For join the middle point of CD to the middle point of AB; then the centre of pressure must be at some point of this straight line, because the pressure on each horizontal strip may be supposed to act at the middle point of the strip. Thus the only question is how far down this straight line the centre of pressure will be; and the answer is two-thirds of the way down, so that its distance from the top will be twice the distance from the bottom. In fact the problem of finding the centre of these pressures is the same as that of finding the centre of gravity of a triangle. For suppose a triangle ABC, such that AB and BC are the same as in Art. 376. Divide this triangle into narrow strips parallel to the base, all of the same width. Then the size of these strips will in A crease in just the same propor

B

portion as their distance from C, that is in just the same proportion as the pressures of the liquid on the successive strips into which we may suppose the side ABCD in Art. 376 to be divided, Hence the weights of the successive strips of the triangle will represent the pressures on the successive strips of the side of the vessel; and thus the centre of pressure will be as far down in the side of the vessel as the centre of gravity is in the triangle; that is two thirds of the way down: see Art. 172.

379. We have hitherto supposed the rectangular side of the vessel to be rertical; but similar considerations apply to the case in which the rectangular side is inclined to the horizon. As in Art. 376 we shall find that the average pressure is that along the middle horizontal line of the rectangle, and is measured by the vertical depth of this straight line below the surface of the liquid. Thus the whole pressure on the side is equal to the weight of a column of the liquid having the side for base, and half the vertical depth of the side for height. The position of the centre of pressure is the same as if the side were vertical.

380. We need not pursue the subject further, but we may state a general result that is obtained by theory. If a plane area of any form is immersed in a liquid the pressure is the same at all points if the area is in a horizontal position; but if the area is not in a horizontal position the pressure is greater as the vertical depth becomes greater. The average pressure is that at the centre of gravity of the plane area. The whole or resultant pressure is equal to the weight of a column of the liquid having the plane area for base, and the vertical depth of the centre of gravity of the plane area for height. No simple rule can be given for determining the position of the centre of pressure.

381. The preceding result admits of a certain extension, which, though of no practical importance, requires notice, for it is sometimes given in books in such a manner as might mislead an incautious reader. Suppose a body having a curved surface, for example a sphere, to be immersed in a liquid. Or suppose a vessel in the form of a curved surface, for example a common bowl, to contain liquid. It is still true that the sum of the pressures of the liquid on the curved surface is equal to the weight of a column of the liquid having this surface for base, and the vertical depth of the centre of gravity of the surface for height. But we must remember that it is the sum of the pressures and not the resultant of them which has this value; the pressures not being all parallel, their sum and their resultant are altogether different. Now there is no special mechanical importance belonging to the sum of a set of forces, though there often is to their resultant: hence this proposition relative to the whole pressure on any curved surface is really of no practical value.

382. One remark may be placed here, which will be of use as we proceed. Suppose a mass of liquid at rest in a vessel, and fix the attention on any definite portion of this mass; the portion may be in the form of a cube or of a sphere, or of any body whatever, regular or irregular. The liquid surrounding it will exert pressures all it, but as the definite portion remains in equilibrium

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