Page:Encyclopædia Britannica, Ninth Edition, v. 12.djvu/459

443 HYDROSTATICS.] HYDROMECHANICS stratum with our approximations being the same in the undisturbed and disturbed states ; and therefore dk M k* Laplace, on the assumption that the cubical elasticity is double tho pressure, and therefore the pressure proportional to the square of the density, has integrated the differential equation for the ellip- ticity, his assumption amounting to putting -^7 -.? = - a 2, a constant. M dk More generally, to make the equation for the ellipticity integrable, we may put 4-n-P dp ~M~ dk~ where a and n are constants, the negative sign being taken because f is negative for stability. This assumption reduces the differential dk equations for p, M, and Me to equations reducible to Eessel s differential equation, and therefore p, M, and e can be expressed by Bessel s functions. Laplace s assumption amounts to putting = 2, and then the Bessel s functions which occur are of the order J, f, and f. Then p = a-^-S, where a- is the density at the centre of the qk earth, and therefore M== (sin qk-qk cos qk), sin a- a cos a (3 3 -57T.-1 ) sin qk, cos qk (pk 1 / qk sin qk qk cos qk where a = qK, and the value of q must be determined from the con dition that the mean density of the earth is twice the density at the surface. Centre of Pressure. When a plane area is exposed to fluid pressure on one side, the resultant force experienced by the . irea is a single force perpendicular to the area, the sum of all the separate pressures, and acts through a definite point called the centre of pressure. If p be the pressure at the point xy, the axes being taken in the plane of the area, then the resultant force intersection with the free surface, with respect to the momenta! ellipse. The centre of pressure of a rectangular area, with a side in the free surface, is at | of the depth of the lower side ; of a triangle with vertex in the free surface and base horizontal is f of the depth of the base ; of a triangle with base in the surface is the depth of the vertex. Mctacentre. We have found from Archimedes s principle the conditions of equilibrium of a floating body, and we must now examine whether the equilibrium is stable or unstable. Let ACB (fig. 7) represent the cross section of a floating body, like a ship, and let G be the C.G. of the body, and II that of the liquid displaced, supposed homogeneous. Let the body be turned through a small angle in the plane of the paper, whose circular measure is 0, so that the volume of liquid displaced remains the same. Then, if W denote the weight of the body, and therefore also of the liquid displaced, the resultant force due to the liquid on the body in the displaced position is a vertical force W acting vertically upwards through H, the new C. G. of the displaced liquid. In order that the volume displaced may remain unaltered, it is necessary that the line of intersection of the two planes of flotation AB and DE should pass through the C.G. of the area of the curve of flotation. For dA. denoting an element of the area A of flota tion, and x its distance from the line of intersection of the planes AB and DE, the element of volume traced out by dA when the body is displaced being OxdA, we must have f0xdA = 0, or fxd A = ; which proves that the line of intersection of the planes of AB and DE passes through the C.G. of the area of flotation. The force W acting upwards through H is therefore equivalent to an equal force W acting upwards through H, and a couple, due to the moment of the weight of AOD upwards and BOE downwards, the moment of which is therefore, in gravitation units, k denoting the radius of gyration of the area A about the line whose projection on the plane is 0. Since HH is the arc of a curve, such that the tangent at H is and, if x, y denote the coordinates of the centre of pressure, ffpdxdy Jfpdxdy The centre of pressure is therefore the C.G. of the plane area, supposed a lamina of vari able density p. If p is uniform, the centre of pressure is obviously the C.G. of the area. For a homogeneous liquid at rest under gravity, p is proportional to the depth below the surface, that is, proportional to the perpen dicular distances from the line of intersection of the plane of the area with the free surface of the liquid. If the equation of this line be x cos a + y sin a-p = Q , ffx(p - x cos a - y sin a)dxdy f/(p - x cos a - y sin a)dxdy - ffy(P -x cos a-y sin a)dxdy ff(P -x con a-y sin a)dxdy If the origin be taken at the centre of gravity of the area, and if the axes be the principal axes of the area, then ffdxdy = A, the area , ffxdxdy =, ffydxdy = , ffxydxdy = f = Aa 2, ffx*dxdy=*AW; then and a and b being the semi-axes of the momenta! ellipse of the area. 1 lierefore 2 i 2. x= - cos o. y sin a ; P P and therefore the centre of pressure is the antipole of the line of parallel to AB, therefore HH is ultimately a straight line perpen dicular to GH, and W. HH or HH -flg = a V ^ if V be the volume of liquid displaced. If H M be drawn vertically upwards to meet HG in M, (hen M is ultimately the centre of curvature of the locus of H in the body, and is called the metacentre, and If M lies above G, the fluid pressure tends to restore the body to its position of equilibrium, and the equilibrium is therefore stable ; but if M lies below G the equilibrium is unstable. Generally we see that, if planes be drawn cutting off constant volumes from a solid, the principal radii of curvature of the locus k of H, the centre of gravity of the volume cut off, will be -~- and -ssr-i where V is the volume cut off, A the area of the cutting plane