ON THE PRESSURE AND EQUILIBRIUM OF FLUIDS Tuts principle, which has been adopted as the founda tion of hydrostatics by Euler, D'Alembert, Bossut, and Prony, is a necessary consequence of the definition which we have already given of fluidity ; for, since the parts of a fluid yield to the smallest pressure, any particle, which is more pressed in one direction than another, would move to the side where the pressure was least, and consequently the equilibrium would be destroyed. If the particles are equally pressed in every direction, it is equally evi dent;that the mass of which they are composed must be in equilibrium.
Although the preceding principle is rigorously true only of perfect fluids, yet, in the case of water, alcohol, &c. where the cohesion of the particles is not very great, the inequality of pressure, under which an equilibrium might exist, must be extremely small ; and it is accordingly found, that the principle is experimentally true in these fluids. For if, at a given depth below the surface of water in a vessel, an aperture be made, and a piston be applied to the aperture to prevent the water from flowing out, it will be found, that the piston will be pressed outwards with the same force, whether the aperture is horizontal or vertical, or inclined at any angle to the horizon.
Cor. If a number of pistons E, F, G, are applied to apertures of different sizes in the sides of a vessel ABCD full of water, the forces with which the pistons arc ap plied will be in equilibrio, if they are proportional to the apertures to which they are applied. Plate CCCXIII. Fig. 1.
Since the pressure of every part of the piston E is transmitted to every part of the piston F, and vice versa, it follows, that these pressures will be in equilibrio, if they arc equal. But the sum of the pressures propagated by E is proportional to the arca of the aperture E, and the sum of the pressures propagated by F proportional to the area of the aperture F ; consequently there must be an equilibrium between these opposing pressures, when E : F=area of E : area of F. The same is true of any num ber of apertures.t When any fluid, influenced by the force of gravity, is in equilibrio in any vessel, its surface is horizontal, or at right angles to the direction of gravity.
Let the surface of the fluid have the curvilineal form A p B, Fig. 2, and let the force of gravity with which any particle p is influenced be represented by the vertical line p o. This force p o may be resolved into the forces p p 7/, coinciding with the elementary portions of the surface on each side of I:. Now, the particle fi being in equili brium, it is pressed equally in every direction ; and, there fore, the equal and opposite forces m p, n p, exerted against p by the neighbouring particles, must be equal to p in, p n; hence the force p m is equal to p n, the angle nip 71 must be bisected by p 0, (See DYNAMICS, Sect. III.) and the elementary portion of the curve must be perpen dicular to p o. As the same is true of every other part of the fluid surface, it follows that this surface must be a horizontal straight line, if the directions of gravity at dif ferent points are considered as parallel, or a portion of a spherical surface, if the directions of gravity meet in one point.
Cor. It follows from this proposition, that the surface of a fluid must be perpendicular to the resultant of all the forces which act upon it. Hence the general surface of the ocean will not be perpendicular to the direction of gravity, but to a line which is the resultant of the action of gravity, of the centrifugal force, and of the attraction of the planetary bodies.
The effect of the centrifugal force, combined with gravi tation, is such, that the surface of the water assumes a parabolic form, as shewn in Fig. 3. When any number of fluids of different densities are put in the same vessel, and are made to revolve round an axis, or if they are put into a glass globe, and turned by the whirling table, their separating surfaces always assume the form of parabolic conoids, when the axis of rotation is vertical.
the surface of the fluid in each branch will be in the same horizontal plane.
Let ABCD, Plate CCCXIII. Fig. 4, be a syphon with three branches, AB, CB, DB, communicating with each other at B. If water is poured into this vessel till it rises to A in one branch, it will rise to the same height in the other branches, so that ADC is a horizontal line perpen dicular to the direction of gravity. Let the syphon be re moved, and let the water which it contained form part of the fluid in the vessel a b c d, in which it has the horizontal surface a ADC b, it is easy to suppose that a portion of the water, of the same form and thickness as the syphon, may be converted into ice, without changing its place or its volume.. The equilibrium of the water is obviously not affected by such a change ; and, therefore, the water will stand at the same height ADC, in a syphon of ice ; and, consequently, the same will happen, whatever be the substance of which the syphon is composed. The same conclusion would have been obtained, by supposing all the water frozen, excepting that portion which was at first in cluded in the syphon.
The arts of levelling and of conducting water are founded upon the preceding proposition. As water will always rise to the same level as the spring from which it flows, it may be conveyed in pipes through the deepest vallies, and over the highest eminences, provided the pipe never rises to a greater height than the source of water. Had the ancients been acquainted with this simple prin ciple, they might have saved the construction of those expensive aqueducts with which their towns were sup plied with water.
Levels are sometimes made upon the principle con tained in the proposition. DIr Keith's mercurial level is nothing more than a syphon filled with mercury, with a float on each branch, which supports two sights. See Edinburgh Transactions, vol. ii. p. 14 ; and our article