whole cone KOL from O beingh and that of the paraboloid KAL from A being th, we have πr2h. h = x (πr2h – πr2h) + ἐπr2h. ( + h), 6h = 2x + 5h, x = sh, a result which shews that the centre of gravity of the solid portion of the cone coincides with the vertex A of the inscribed paraboloid. y Again, the equation to the generating parabola KAL being 2r2 万x, = 4 V. HM = πη* = But AH is equal to x; hence 2πγ 4 πγ x2, h2 πνχα Sxdx = h and therefore, OA being equal to th and OH' to x', Also, the weight of the fluid displaced being equal to the sum of the weights of the solid portion of the cone and the fluid poured into the paraboloid, V. σ + (vol. of cone vol. of KAL) ρ = V'. σ', The value of x is determined by the two equations (1) and (2). CHAPTER IX. TENSION OF VESSELS CONTAINING FLUID. LET p be the pressure exerted at any point of a vessel of any form by the fluid which it contains; r, s, its principal radii of curvature at the point; and u, v, the respective tensions of the material of the vessel in the corresponding principal sections. Then p = If the vessel is immersed in fluid, p must be here taken to represent the difference of the pressures of the exterior and interior fluids. If the tensions be the same in both principal sections, u, v, being each denoted by the same letter t, we have Supposing the vessel to be cylindrical, one of the principal radii of curvature will be infinite: putting s = ∞, we have a relation between the pressure, the tension, and the radius of curvature, the discovery of which is due to James Bernoulli. See Jacobi Bernoulli Opera, tom. II. p. 1043, de Curvatura Fili ab innumeris potentiis extensi. 1. A tube, every horizontal section of which is circular, is filled with fluid which is acted on by gravity, the axis of the tube being vertical: to find the law of the thickness of the tube that it may be, in regard to the stress of the fluid, of the same strength throughout. Let t denote the tension of any circular section of the bore of which r is the radius, and p the pressure of the fluid at each point of this section. Then Let h represent the depth of this section below the surface of the fluid, g the force of gravity, p the density of the fluid: then and therefore, from (1), p = gph, t = gphr..................(2). Suppose T to denote the thickness of the tube at the section, and f the force exerted by a unit of thickness, which, by the condition of the problem, will be the same throughout the tube. Then t = fr, and therefore, from (2), fr = gphr, τα hr. If the tube be of uniform bore, its thickness must vary from section to section as the depth below the surface of the fluid. Encycl. Metrop. Mixed Sc. vol. 1. p. 177. 2. A given mass of known elastic fluid is confined in a slightly elastic spherical envelop of given material, the natural radius of which is assigned. Supposing the tension of the envelop to be as the linear extension, to determine approximately the increment of its radius. Let r be the radius of the expanded envelop, R its natural radius, p the pressure of the fluid at each of its points. Then, t being the tension, (1). Let u denote the modulus of elasticity of the material of the envelop; then r R = pRt................(2). From (1) and (2), since u is, by the condition of problem, supposed to be small, we have r - R = Rp...(3). Let k represent the elasticity of the fluid, supposing a unit of its mass to be compressed into a unit of volume; then, π being the volume of the envelop, and M being taken to represent the mass of the fluid in the envelop, 3. A paraboloid is filled with fluid, the pressure at any point of which is equal to p; to determine the tension at this point, its distance from the focus being r, the tension in both principal sections being supposed to be the same. The radius of curvature of the generating curve is equal to and the radius of curvature of the other principal section, y2 = 4mx being the equation to the generating parabola, will be, if 0 be the angle between the ordinate and the normal, Hence r', s', denoting the two radii of curvature, 4. To find the form of a rectangle of cloth which, having two opposite sides supported parallel to each other in a horizontal plane, is pressed by the weight of a fluid contained in it, the fluid being supposed to be prevented running out from the |