HYDROSTATICS. 384 HYDROSTATICS. difTerent areas, A, and A„ the forces which must lie iipplieil to these pistons from without to pre- vent the lluld from pressiiij; them outward arc PA, and I'A. — omittinj^ any action of {;ravity. Therefore, if A, is small compared with A„ the force on the former piston is small coni])ared with the balancing force on the latter; so a small force may produce a large one. In a liquid which 18 almost inconipressihle, I' may he very great ; and so the force produced may he enormous ; but in a gas. which is easily compressed. P is never very large: and so the force produced is small. (This principle is that of the hydrostatic press. See HvDR.ii.ic Press.) The total pressure, i.e. P + pnh, at all points in the same horizontal level in any one fluid, regardless of the shape or size of the containing vessel, is the same: tor imagine a vessel with a liorizontal bottom, all i)oint-s of the lluid along this must have the same pressure, otherwise the fluid would How: the |)ressure at any )ioint at a level h centimeters above this bottom plane is less than that at the bottom by an amount pph — the .same for all points in the plane. If, now, portions of the vessel are imagined removed, so as to leave a vessel of any shape, the pressure at the various points remains imatrected. The pres- sure on any portion of the containing walls va- ries with the depth; so the thrust outward on this portioit is the resultant of a series of par- allel forces, increasing from top to bottom: the point of application of this resultant is called the 'centre of pressure.' If the wall is a vertical rectangle, the centre of the pressure, due to grav- ity, for a liquid, is at a distance one-third the height from the bottom, etc. Arciiimedes's Pri.nxiple. If a solid is com- pletely immersed in a fluid, the latter exerts on it pressures from all directions perpendicular to the elementary portions of the surface of the solid. These pressures will produce an upward force on the solid equal to the weight of the fluid displaced by it; because, if the space occupied by the solid were to be fdled with the fluid, there would be equilibrium; and. as this substitution has not changed the pressures at the boundary surface of the vidume formerly occupied by the solid, the resultant of these surface pressures must just balance the weight downward of the fluid. This buoyant force acting on the solid must then equal the weight of the fluid displaced by the .solid and must have a vertical line of action passing through the centre of gravity of the displaced fluid. This is called 'Archimedes's principle.' The apparent loss in weight of a bo<ly immersed in water, the suspension of a balloon, etc.. are illustrations of it. It evidently gives a method for comparing the density of a solid with that of a liquid: for the weight of a solid in a vacuum is pgv, if p is the density of the solid and r its vohime; and its apparent loss in weight when weighed suspended in the liquid is ynr. if p' is the density of the liquid; both the weight and the loss in weight may be measured, and thus the ratio of p and p' deter- mined. Free SrRF.rE W.wts. Liquids diflTer from ga.ses in having definite volumes; consequently, as noted before, if a liquid is poured into an open vessel, it will have a surface in contact with the atmosphere. This is called the 'free surface.' If the liquid is in equilibrium, this surface must be at right angles to the forces acting ou it. Thus a liquid at rest in an open vessel has a horizontal surface owing to the earth's force of gravity. If this surface is disturbed, e.g. by (Irojiping in a stone, the displaced portion will return to its previous level owing to the action of gravity, but its inertia will cau.se it to con- tinue its motion, making a ilisplacement in the opposite direction; then it will return, etc., thus vibrating up and down and producing waves out over the surface. These gravitational surface waves on a liipiid advance with a velocity which depends upon the nature and depth of the liquid and the wavelength of the waves. Thus, if the liquid is shallow comjiared with the wavelength, the velocity is given by the formula v^=Vhg, where It is the depth of the liquid and g is the acceleration of a falling body. If the liquid is deep compared with the wave-length, the velocity of the waves is where X is the wave-length of any train of waves, and ir = 3.1410. If the waves are ex- tremely short — i.e. ripples — their motion is not due to" gravitation, but to capillarity (q.v.), be- cause the surface is increased in area by the ripples; and the capillary forces are, in this case, large compared with the gravitation ones, the surface being so slightly elevated or de- pressed. The velocity of these capillary waves is y p where T is the 'surface-tension' of the liquid. (Sec Capillarity.) If the vessel containing the liquid is cylindrical, and if this is set rotating rapidly about its a.xis, the free surface will 'no longer be horizontal, becau.sc now the surface is acted upon by both gravity and 'centrifugal force.' The shape of the surface will be such as to be at right angles to their resultant, and will actually be a paraboloid of revolution. If a liquid is poured into a vessel which con- sists of several parts, the liquid will stand at the same height in all parts provided they are wide and open to the atmosphere; for, as has been proved, all points at the same horizontal level in
- >, fluid are at the same pressure, and conversely;
and the points in the free surfaces are at the same pressure, viz. that of the atmosphere. If, however, a liquid'is poured into a U-tube which stands erect, and then a lighter fluid, which does i.ot mix with the first, is poured into one arm, the level of the surfaces of the liquids in the two arms is not the same. The pressure at points which lie in the horizontal |(lane passing through the surface of separation of the two liquids must bo the same, because they are all connected by one liquid. The pressure at the points in one arm in this level is p,ff/i, + P, where p, is the density of the liquid in that arm, /i, is the verti- cal height of its free surface above the level jdane, and P is the pressure of the atmosphere on the free surface; similarly the pressure at this level in the other arm is p^nh. + P, where p. is the density of the liquid whose vertical height from the level to the free surface is A,. There- fore P,gh, + -P=p.j,h,+ V or p./i, = p.h. This principle evidently furnishes a method for
