444 HYDKO MECHANICS [HYDROSTATICS, intercepted by the surfaces, and& 1; 2 the principal radii of gyration of this area about the centre of gravity of the area. The vertical through H will intersect HG only when the plane of displacement is a plane of symmetry, that is, if it is perpendicular to k 1 or k. 2 . Generally for finite displacements in a plane the locus of M will be the evolute of the curve which is the intersection of the plane with the locus of H. The surface the locus of H is called the surface of buoyancy, and the surface which is the envelope of the planes of flotation is called the surface of flotation. If 7-j, r., be the principal radii of curvature of the surface of buoy ancy, then r, =A^L_Ii T _A^!_li V V a V V If HI, Ko be the principal radii of gyration of the surface of flota tion, Mons. E. Leclert has proved that R,-^!l- r + I^i R -^h-r +L* i 1 dV * dV z ~dV~ 2 dV For small oscillations, if we suppose the liquid pressure the same as if the liquid were at rest, the body oscillates as if the surface of buoyancy moved upon a horizontal plane. N Fi Next, suppose a body completely immersed in heterogeneous liquid, which must for equilibrium be arranged in horizontal strata of equal density, and suppose p=f(z) the density of the liquid at any depth z ; let G be the centre of gravity of the body and H that of the liquid displaced (fig. 8). When the body is in equilibrium, we must have G and H in the same vertical line, and fffpdxdycb-W, the weight of the liquid displaced, using gravitation units of force again. Suppose the axes of coordinates fixed in the body, and take GM as the axis of z ; suppose the body turned through a small angle 6 about the axis Oy, and let H be the centre of gravity of the dis placed liquid in the new position. The density of the liquid dis placed at a point P, whose coordinates are x, y, z, is now /(NP)=/(z cos 6 + x sin 6) neglecting e 2 ; and, to the same order of approximation, /YY* I , X /(-) + 8rf (~ 1U1 == JU (7V 1 Drawings of the curves of buoyancy and of flotation, and of the metacentric curves, are given in the plates of illustration of a paper on the " Calculation of the Stability of Ships." by W. II. White and W. John, read before the Institution Of Naval Architects, March 1871. since fffxf(z}dxdydz=Q, H lying in the axis of s. If the vertical through H meet 0~ in M, M is called the meta- centre, and is the centre of curvature of the locus of H, and -. //~ , tfr/i W W if A& 2 denote the moment of inertia of a horizontal plane section of the body at a depth 2, about the line of intersection with the plane of yz. We have here supposed that H lies in the plane of x~ ; but this will only be true for two directions of displacement. In general for any displacement, if x, y, z be the coordinates of H , y W JffsyfWxdydz W and therefore y = 0, only when the axis of y is a principal axis of the body, supposed of density - . ctz When p is discontinuous, as in the case of the body floating in homogeneous fluid, then the integral fA.k 2 dp will have a term AJc^p, due to the discontinuity at the surface, and the rest of the integral will vanish, because _ = 0. dz For a body floating wholly immersed in two liquids, the upper of uniform density p and the lower of uniform density p, HM = ^ Tension of Flexible Surfaces exposed to Pressure. In hydrostatics it is usual to determine the circumferential and longitudinal tension produced in a thin circular cylinder, due to- uniform internal pressure, and also to determine the tension of a spherical surface, like a soap-bubble, due to the excess of the inter nal pressure over the atmospheric pressure. Let r be the internal radius of the cylinder, e the thickness (sup posed small), p the internal pressure, and t,t tQ circumferential and longitudinal tension per unit of area caused by the pressure p. If we suppose the cylinder divided into two halves by a diametral plane, and consider the equilibrium of unit length of either half under the resultant of the fluid pressure over the half-cylinder and the tensions at the ends of the diameters, the resultant of the ten sions must balance the resultant of the fluid pressures, which is the resultant pressure on the diametral plane, since the resultant of the uniform pressure on a closed surface is zero. Therefore t r or = p c To determine the longitudinal tension t , consider that the resultant pressure on the end of the cylinder, which is ^.irr 2 , is balanced by the resultant of the tensions round a circumferential seam, which is t c.2irr ; and therefore Znt cr wpr^ ; and therefore t = I. Thus in a boiler, half an inch thick, and 3 feet in diameter. a pressure of 150 lb to the square inch makes ^ = 5400, Z = 2700. For a sphere of internal radius r, and small thickness c, supposing it divided by a diametral plane, then the resultant tension round the circumference, tc. 2irr, must balance the resultant fluid pressure p.Ttr 2 , supposing p the excess of the internal over the external pres sure ; and therefore In the experiment with the Magdeburg hemispheres, where two hemispheres were joined by an air-tight joint and the air say half exhausted, then, with a pound and inch as units, p = 7 5, suppos ing 15 the atmospheric pressure ; and if the diameter of the hemi spheres be 3 feet, then r = 18 ; and the force required to separate the hemispheres would be The tension of flexible surfaces is considered more fully in the
article CAI-ILLAIIY ACTION.
