372. The theoretical demonstration of the statement is so simple that it well deserves the little attention which is necessary in order to understand it.

A P

S

R

B

Let P be any point in the base of a vessel containing liquid: we wish to shew that the pressure on an assigned area at P is equal to the weight of a column of the liquid which would stand on that area, and reach up to the surface of the liquid CD. If a vertical straight line can be drawn in the liquid from P to the open surface the proposition has been already established, namely in Art. 359; the case which we have to examine is that in which this vertical straight line cannot be drawn in the liquid owing to the inclined sides of the vessel. In this case however it will be possible to pass from the point P to the open surface by a zigzag composed of vertical and horizontal straight lines; thus in the diagram we have PQ and RS vertical, and QR horizontal.

Now in the first place the pressure at R is known by Art. 359; it is proportional to the depth RS.

Next we shall shew that the pressure at Q is the same as the pressure at R. For suppose the liquid in the form of a slender horizontal rod along QR, with parallel vertical ends, to become solid; the pressures on its ends are the only horizontal forces acting along the rod, and these must therefore be equal for equilibrium.

Finally the pressure at Q being equal to the pressure at R the column of liquid PQ is in precisely the same circumstances as it would be if it were placed vertically under RS instead of in the position it occupies. Hence the pressure on the assigned area at P is precisely the same as it would be if a vertical straight line could be drawn in the liquid from P to the open surface.

373. Thus the pressure of the liquid on the base of any vessel, which is open at the top, is equal to the weight of such a column of the liquid as would stand vertically

over the base, and reach up to the open surface. The pressure may be supposed to act at the centre of gravity of the base: see Art. 172.

XXIX. PRESSURES ON THE SIDES OF VESSELS.

374. We have now sufficiently considered the pressure on the base of a vessel containing liquid; we proceed to the pressure on the sides. The fact that the pressure increases as the depth increases suggests an obvious practical remark with respect to constructions which are intended to resist the pressure of liquids. Suppose we have to carry a canal across a low valley, so that is necessary to make embankments to serve as artificial sides for the canal. Since the pressure of the liquid increases in the same proportion as the depth, the strength of the embankment ought also to increase with the depth: thus the embankment should be wide at the bottom, and may become gradually thinner towards the top.

375. Again, the pressure in a liquid depends on the depth but not at all on the length of the vessel in which it is contained. Hence if the water of a pond or canal is to be restrained at one end by a flood-gate or dam, it will not matter whether the channel of water is a few yards or a mile long, so far as the flood-gate or dam is concerned; the pressure is the same on it in the two cases. This is a fact which often seems very puzzling to persons who have not attended to natural philosophy; they do not consider that when the channel is lengthened so as to involve more water the sides are also lengthened which confine it, so that there is no necessary increase of pressure on the end. It must be remembered however that the statement assumes the water to be at rest: if the water is liable to be thrown into commotion by the wind or other causes it is plain that a large mass of water will in general produce more impression on the restraints than a small mass.

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.

C

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 vertical; 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 over it, but as the definite portion remains in equilibrium