(21) A cup whose outside surface is a paraboloid of revolution of latus-rectum 7, and whose thickness measured horizontally is the same at every point and very small compared with 7, has a circular rim at a height h above the vertex, and rests on the highest point of a sphere of radius r. If water be now poured in until its surface cuts the axis of the cup at a distance 3h from the vertex, and if the weight of water be four times that of the cup, the equilibrium will be stable, if (22) Prove that if a thin conical vessel of vertical angle 2a and weight W, whose c.G. is at a distance h from the vertex, is resting upright in a horizontal circular hole of radius c, it will become unstable when a weight P of liquid is poured into it to a depth x, so as to make ¿Px−2(P+W)c cot a+ Wh cos2a positive. (23) A cylindrical vessel, floating upright in neutral equilibrium, will really be stable if the radius of curvature at the water line of the vertical cross section is greater than the normal cut off by the medial plane. (24) Prove that the metacentric height given by (P/W)b cosec (§ 94) can be made correct to the second order for the ship, when P is removed, by adding to it (§ 112) (P/W)(c—r1), where r1 denotes the radius of the curve of flotation, and c the height of P above the water line. CHAPTER V. EQUILIBRIUM OF FLOATING BODIES OF REGULAR FORM AND OF BODIES PARTLY SUPPORTED. OSCILLATIONS OF FLOATING BODIES. 126. The Equilibrium of a floating Cylinder, Cone, Paraboloid, Ellipsoid, Hyperboloid, etc. When the body has the shape of one of these regular mathematical forms, the curves of flotation F and of its evolute C, of buoyancy B, and of the prometacentres M, or the metacentric evolute, can be determined by various theorems introducing interesting geometrical applications of the properties of these curves and surfaces. For a prismatic or cylindrical body like a log, floating horizontally in water, the various surfaces are cylindrical and we need only consider their curves of cross section. If the section is an ellipse, these curves of flotation and of buoyancy are also ellipses; and the determination of the position of equilibrium will depend on the problem of drawing normals from the C.G. of the body to the ellipse of buoyancy, or tangents to its metacentric evolute; and two or four normals or tangents can be drawn according as the C.G lies outside or inside this evolute. If the sides of the log in the neighbourhood of the water line are parallel planes, the curve of flotation reduces to a point, and the curve of buoyancy becomes a parabola (§ 103). If the submerged portion of the log is triangular, or more generally if the log is polygonal or if the sides in the neighbourhood of the water line are intersecting planes, the curves of flotation and of buoyancy are similar hyperbolas of which the cross section of these planes are the asymptotes. When the cross section of the log is rectangular or triangular, the curves of flotation and of buoyancy are composed of parabolic and hyperbolic arcs, interesting figures of which, by Messrs. White and John, will be found in the Trans. Inst. Naval Architects, March, 1871; also by M. Daymard, I.N.A., 1884. If the outside shape of the body is an ellipsoid or other quadric surface, then according to well-known theorems the surfaces of flotation and buoyancy are similar coaxial surfaces; just as in the sphere, from which the ellipsoid may be produced by homogeneous strain. If the surface of the body is a quadric cone, the surfaces of flotation and of buoyancy will be portions of hyperboloids of two sheets, asymptotic to the cone. OF THE UPRIGHT CYLINDER. 127. The Cylinder, floating upright. 191 When a cylinder of S.G. s floats in water, the surface of flotation F reduces to a fixed point on the axis, so long as an end plane of the cylinder does not cut the surface of the water, and the surface of buoyancy is a paraboloid (§ 103). If h denotes the height of the cylinder and x the length of axis immersed, then x=sh; and for displacements in a vertical plane from the upright position of equilibrium the curve of buoyancy is a parabola (fig. 44), and BM=Ak2|V=k2/x=k2/sh. The equilibrium is stable in the upright position if the C.G. G lies below M, or if BM > BG, or the upright position of the cylinder is unstable in the corresponding vertical plane of displacement, and the cylinder "lolls" to one side. In this case the greatest value of k2/h2 is, and then s; so that the cylinder will float upright in any liquid if h2 = 8k2. When the horizontal cross section of the cylinder is a rectangle of breadth b, k2= b2 (§ 40); and this prismatic log cannot float upright, if 12 But if b2/h2 > 3, b/h > 1/6, the log will float upright in any liquid. In a log of square vertical section bh; so that it cannot float with faces horizontal and vertical, if {+}√3(=0·79) > s > 1− } /3(=0·21), When the cross section of the cylinder is a circle of radius a, 12=4a2 (§ 40); and this cylinder cannot float upright, if b+b √(1−245) > 8 >b-b√(1–2) h2 But if h/a</2, this cylinder will always float upright, like a bung. As an exercise, the student may prove that the body in fig. 44, if floating in two liquids of S.G.'s 81 and 82 will be in stable equilibrium in the upright position if If the body comes to rest when floating in water with its axis at an inclination 0, then m must coincide with G. But the curve BB2 being a parabola (§ 103), k2 k2 Om=OB+BM+Mm=x+-+ tan20; and therefore, if Om=OG, x2-hx+2k2+k2tan20=0. Thus, for instance, if fig. 44 represents the cross section. of a rectangular log of breadth b, and if E just reaches the surface of the water in the position of equilibrium, |