Suppose that a force is applied to the piston at G, just sufficient to keep it in its place, so that the liquid remains in equilibrium. Let now the piston at E be pushed down with any force, say a force of one pound; it will be found, as we said before, that to preserve equilibrium the piston at F must be pushed down with a force of one pound: and moreover we must push in the piston at G with a force of one pound in addition to the force already exerted on it. Thus the force applied at E is transmitted to the equal area at G. Also if the area of the piston at G is ten times the area of the piston at E, then when a force of one pound is applied to the piston at E we must in order to preserve equilibrium apply a force of ten pounds to the piston at G, in addition to the force which it was necessary to exert to keep this piston from being thrust out before any force was applied to the piston at E. 355. In our illustration we have supposed the tubes at E and F to be vertical, and that at G to be horizontal; but the principle is not to be restricted to these cases. The side of the vessel in which the tube is supposed to be inserted need not be necessarily either horizontal or vertical, but may be inclined at any angle to the horizon. Still the result will hold, namely, that when equilibrium has been obtained by applying proper forces to the pistons, then if any additional force be applied to one piston we must apply an equal additional force to every portion of the same area in all the other pistons, in order to maintain equilibrium. 356. The principle of the transmissibility of pressure through a fluid explains the action of a little contrivance which is called the hydrostatic paradox; the name is given because at first sight the effects seem out of proportion to the causes in action. CD and EF are flat boards, which are connected by flexible leather, or cloth, so as to form a water-tight vessel. AB is a vertical tube which communicates with the vessel. Let the vessel and a part of the tube be filled with water, and suppose a piston to work in the tube and to be retained in its place by a suitable force. Suppose, for an example, that the area of the bore of the tube is one square inch, and that the area of the upper board of EF is a thousand square inches. Then if the piston is pushed down with an additional force of one pound the board EF will be thrust upwards with a force of a thousand pounds; so that in fact the board would support the weight of a thousand pounds placed on it without sinking down. It will be seen after reading the next Chapter that instead of using a piston in the tube AB the required force may be obtained by making the column of water in the tube of sufficient height. We shall see hereafter that the principle of the hydrostatic paradox is the essential part of a valu- . able machine called Bramah's Press. The important principle of Art. 208 applies here, namely that what is gained in power is lost in speed: for if we were to force the piston in the tube down through one inch the board EF would ascend through only one thousandth of an inch. XXVII. PRESSURE FROM THE WEIGHT OF LIQUIDS. 357. We have hitherto considered liquids as contained in closed vessels and transmitting to all points any pressure which may be applied at their surfaces; but we have now to treat of the pressure produced by the weight of liquids. 358. Suppose liquid put into a vessel open at the top; then the upper surface will be a horizontal plane. That it is a plane surface is obvious from common observation. To say that it is a horizontal plane means that it is at right angles to the direction of gravity, and this may be established by an easy experiment. If a plumb line be hung over the surface of a liquid at rest the eye can discern that the direction of the plumb-line and the direction of its image reflected in the liquid seem to fall in the same straight line; and when the student is acquainted with the elements of Optics he will know that this shews the surface of the liquid to be at right angles to the direction of the plumb line. The result may also be established by reasoning. Suppose the surface of a liquid to be curved, as denoted by ABCDE. Consider a portion of the fluid BCD such as would be cut off by a plane BD inclined to the horizon. Then this portion would be like a body placed on a smooth Inclined Plane, acted on only by its own weight; and so it would not be in equi B librium but would run down the Plane. ୯ D E 359. Let there be an open vessel with vertical sides containing liquid. Consider any portion of the area of the base, as for instance, one square inch near the point P; then the liquid itself produces on this square inch a pressure equal to the weight of a column of the liquid, of which the area of the base is one square inch, and the height is the vertical depth of P below the surface of the liquid Thus if the depth PQ is 28 inches, and the liquid is water, the pressure on a square inch of the base at P would be Now a equal to the weight of a column of water of which the base is one square inch and the height is 28 inches. Such a column would contain 28 cubic inches of water. cubic foot of water weighs about 1000 ounces Avoirdupois, ounces, and 28 cubic inches so that a cubic inch weighs 1000 1728 ounces, that is about a pound. It will be con weigh venient to remember that a column of water of which the area of the base is a square inch and the height is 28 inches weighs about a pound Avoirdupois. 360. But let us advert to the evidence for the truth of the preceding statement. We might contrive some experimental test. For instance the vessel might be placed high on supports at its corners, so as to allow of easy access to the base; then a tube might be inserted at P in which a piston should work; and the force necessary to sustain this piston in its place could be found by trial. Or we might adopt some methods of reasoning. For instance the sides of the vessel being vertical it seems obvious first that the whole pressure on the base must be equal to the whole weight of the liquid, and next that the pressure on any assigned part of the base will be proportional to the area of the part; and from these two natural suppositions the result will follow. There is also a method of reasoning which may appear somewhat artificial to the reader at first, but which well deserves attention as it is very useful in the theoretical investigations of the subject. Consider a vertical column of the liquid which has for its base an area of a square inch at P, and reaches up to the surface of the liquid. Conceive this to become solid; then we may take it as obvious that the pressure on the square inch is not altered. The weight of this solid column must be supported by the resistance of the base, which is equal and opposite to the pressure the liquid exerted on the base. For the liquid around the column will exert pressures on it only in horizontal directions, and so will in no degree counteract the weight of the column. Thus finally the pressure on the square inch of area at P is equal to the weight of the column of liquid standing on this square inch of area as base. 361. Next suppose a plane area of one square inch to be placed at any point between P and Q, in a horizontal position. The pressure on one side of it, say the upper side, will be equal to the weight of the column of liquid above it. This will appear obvious on reflection. We might suppose all the liquid below the plane area to become solid, and allow that the pressure on the plane area would remain unchanged: then this case reduces to the former. 362. Next suppose that at any point of the liquid we put an area of one square inch inclined to the horizon. The pressure on one side will be the same in amount as if the area were horizontal and at the same depth. The words at the same depth are used for brevity; they require a little explanation in order to bring out their strict sense. The depth of the inclined plane must be understood to mean the average depth, that is the depth of the centre of gravity. It would not be easy to obtain a very simple direct verification of this statement; but we may give an experimental illustration which will serve to render the meaning clear. Let there be a flat piston moving in a tube closed at the bottom and quite water-tight; and in the tube let there be a spring which resists the motion of the piston, so that a certain pressure must be exerted on the piston to maintain it at a certain position in the tube. Put the whole under the surface of the liquid; then the pressure exerted by the liquid pushes the piston in until there is equilibrium between the pressure and the resistance of the spring. Then for all positions of the piston so long as the centre of gravity of its area remains at the same depth the piston will remain in equilibrium. 363. All the results we have given in this Chapter are obtained on the supposition that the upper surface |