of the liquid is left free. If a lid is put on the upper surface, and pushed down, this gives rise to an additional pressure which is transmitted to every point of the liquid. It will appear hereafter that the atmosphere produces a pressure of about fifteen pounds on every square inch of surface exposed to it; and this pressure is transmitted through the liquid to a square inch of area placed in any position within the liquid.

364. But at present we leave out of consideration the action of any other force except the weight of the liquid itself; and the results at which we have arrived may be summed up briefly thus: the pressure at any point of a liquid is proportional to the depth of the point below the surface, and is the same in every direction. Like many other brief statements this would be scarcely intelligible without previous explanations. We measure pressure at any point by the pressure on a certain small area, say a square inch, so placed as to have its centre of gravity at the point; and when we say that the pressure is the same in every direction we mean that this area may be placed at any inclination to the horizon.

365. The fact that the pressure is the same in all directions round any assigned point, to which we have just drawn attention, is quite distinct from the fact that liquids transmit pressure from one point to another: both are very important properties of liquids.

366. The reader will observe that we speak of the pressure of a liquid at a point and not of the pressure on a point; in order to form a notion of the pressure of a liquid we must suppose that it is exerted on some definite area; this area may be very small, but it is not what is called a point in geometry.

XXVIII. VESSELS OF ANY FORM.

367. In the preceding Chapter we supposed liquid to be contained in a vessel with vertical sides; but we must now proceed to some other cases.

Let us suppose liquid to be contained in vessels which have sides that are not vertical; these sides may slope outwards as in the left-hand side diagram, or inwards

D

A

A

B

as in the right-hand side diagram. The two results which were obtained in the preceding Chapter, and summed up in Art. 364 are still true, and thus we shall be led to some curious and important consequences.

368. Consider the case represented by the left-hand side diagram. The pressure on the base of the vessel is equal to the weight of such a column of the liquid as would stand vertically over the base; thus it is less than the weight of all the liquid contained in the vessel. The weight of the liquid contained in the vessel is equal to the vertical component of the pressure on the vessel; but this does not fall entirely on the base; part falls on the inclined sides. Next consider the case represented by the right-hand side diagram. The pressure on the base of the vessel is equal to the weight of such a column of the liquid as would stand vertically over the base; thus it is greater than the weight of all the liquid contained in the vessel. In this case, as in the former, there is pressure by the liquid on all the vessel in contact with it, and therefore resistance from the vessel on the liquid. But in this case the vertical component of the resistance from the inclined sides tends downwards; and the difference between this and the resistance of the base upwards is equal to the weight of the liquid in the vessel.

369. The following is the general result. Let there be a series of vessels all having flat bases of the same

area, all open at the top, and filled with the same liquid up to the same height; then the pressure on the base of any vessel will be the same, namely the weight of such a column of the liquid as would stand vertically over the base. The vessels may have any shape whatever; they may be like cups, or jugs, or decanters, or pails; and the opening at the top may be as small as we please. It is plain that we have thus a fact of the same nature as that involved in the Hydrostatic Paradox. Suppose that the vessel is in the form suggested by the diagram, large and shallow, with a tall slender neck. Pour in liquid until it fills all the shallow part and the neck up to CD. Then the pressure

A

C

B

on the base of the vessel, represented by AB, is equal to the weight of such a column of the liquid as would stand on this base and reach up to CD: and it is obvious that this may be many times as large as the weight of all the liquid contained in the vessel.

370. The reader will observe that the pressure about which we are speaking is the pressure of the liquid on that. side of the base with which it is in contact, and not the pressure between the other side of the base and the table or ground on which the vessel may be supposed to stand. The latter is equal to the sum of the weights of the vessel and the liquid which it contains, by the ordinary principles of mechanics.

371. Experimental evidence can be furnished of the truth of the general statement of Art. 369. Vessels are constructed of various shapes, as suggested in that Article, and having bases of the same area. These bases are not fixed to the sides of the vessels, but are kept in contact with them by forces which can be exerted by means of a lever. Then it is found that when the vessels are filled up to the same height the same force must be exerted in every case in order to keep the moveable base in its place.

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.

Let P be any point in the base of a vessel containing liquid: we wish to shew that

A P

R

B

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.