47 HYDROSTATICS. THE few explanatory articles and paragraphs that are enclosed in brackets form no part of the Course required to be read by the Candidate for the degree of B.A. Those in smaller type are illustrative of the Definitions and Propositions which they immediately follow. The whole of the extra matter so introduced forms but a very small, and a very easy, portion of the book; and they who come to this subject for the first time will do well not to omit it. 1. CHAPTER I. HYDROSTATICS investigates the conditions fulfilled by the pressures produced by fluids when in a state of rest. 2. DEF. A FLUID is a material body which can be divided in any direction, and the parts of which can be moved among one another by any force however small. [Water, Air, Gas, Mercury, Steam, are all instances of fluids.] 3. FLUIDS have been divided into Elastic, and Non-elastic. (1) ELASTIC FLUIDS are those of which the dimensions are increased or diminished when the pressure upon them is increased or diminished. [Air is an elastic fluid, as will hereafter be proved. So also is gas, and steam.] (2) NON-ELASTIC FLUIDS are those of which the dimensions are not diminished by the addition of pressure, or increased by the withdrawal of pressure. [Water, Mercury, and probably all other liquids, are compressible, but in a small degree only. The resistance, however, which they offer to compression is so great, that the conclusions obtained on the supposition of their being entirely incompressible are free from any sensible error, except in a very few cases.] CHAPTER II. PRESSURE OF NON-ELASTIC FLUIDS. 4. PROP. 1. FLUIDS press equally in all directions. со B Let a close vessel of any shape be filled with fluid, and let A, B, C and D, similar and equal portions of the sides of the vessel, be removed, and their places supplied by plugs fitting the orifices exactly and acted upon by pressures just sufficient to keep them at rest. Then if an additional force P be applied to any one of the plugs, it is found that an equal additional pressure must be applied to every other of the plugs in order to prevent the fluid bursting out. This experiment proves that a pressure communicated to the surface of a fluid at rest is transmitted by the fluid, undiminished and unimpaired, to every other equal and similar portion of the surface the fluid is bounded by. Also, whatever be the directions in which any of these plugs are inserted through the sides of the vessel, the same quantity of pressure is required to keep them at rest; which shews that, at the same point in a fluid, the fluid presses with equal force in all directions. Wherefore, "Fluids press equally in all directions." Q. E. D. 5. PROP. II. The pressure upon any particle of a fluid of uniform density is proportional to its depth below the surface of the fluid. OD MN Let the vertical lines PM and QN be drawn through P and Q two particles situated in the A interior of a fluid of uniform density at rest, and let them meet the surface of the fluid in the points M and N. •RP. Supposing the pressures on P and Q to be produced by the weight of fluid particles of the same magnitude as P and Q placed touching each other along the lines PM and QN, then, since the fluid is of uniform density, the weights of such lines of particles will be proportional to their lengths. .. Pressure on the particle P : pressure on the particle Q :: weight of the line of particles PM :: PM: QN. Again; the fluid being at rest, the pressure of any part of it will not be affected if we suppose the particles (which are at rest) of the fluid upon which the pressure acts, to become connected with one another. Whence it will follow, that whether there be an uninterrupted line of fluid particles (as RO) reaching from a particle R to the surface of the fluid, or part of the line be cut off by a portion ADD of the fluid becoming solid, the pressure upon the particle R will still be the same. Wherefore, "The pressure, &c." Q. E. D. 6. PROP. III. When a fluid is at rest its surface is horizontal. [The manner in which the inatter composing non-elastic fluids acts is twofold; first, by the particles pressing with their weights on those immediately below them; and second, by the property these particles possess of communicating, in all directions and without diminution, any pressure to which they are subjected.] a b Imagine two contiguous particles P and Q situated in the same horizontal plane in a fluid at rest. Since the action of gravity on these particles is perpendicular to the plane in which they lie, its action on either of them can produce no effect in that plane. P and Q are therefore kept at rest entirely by the action of the surrounding fluid, and this action must be the same on each, or motion would ensue. But since fluids press equally in all directions, and the horizontal pressures of P and Q on one another are equal, their vertical pressures, which are as their distances Pa and Qb below the surface of the fluid, must also be equal; and .. Pa = Qb; .. ab is parallel to PQ, and is .. horizontal. And since in like manner the line joining any two adjacent particles in the surface of the fluid may be shewn to be horizontal, therefore the surface itself is horizontal. Wherefore, "When a fluid, &c." Q. E. D. 7. PROP. IV. If a vessel, the bottom of which is horizontal and the sides vertical, be filled with fluid, the pressure on the bottom will be equal to the weight of the fluid. The sides of the vessel being vertical, and the whole of the fluid being conceived to be made up of vertical straight lines of fluid particles, each of these lines will press vertically with its weight, and the sum of these vertical pressures will be the weight of the fluid in the vessel. Now the base of the vessel, being horizontal, will counterbalance and entirely destroy all the vertical pressures upon it. The pressure, therefore, sustained by the horizontal base of a vessel whose sides are vertical is equal to the weight of the fluid in the vessel. Q. E. D. 8. [From Propositions II and IV the following conclusion may be drawn. The pressure exercised by a fluid on any horizontal plane placed in it, is equal to the weight of a column of the fluid whose base is the area of the plane, and whose height is the depth of the plane below the horizontal surface of the fluid. |