If R denote the resistance in the former case and R' in the latter, then, D denoting the diameter of the circle and C the length of the chord of the segment, James Bernoulli: Act. Erudit. Lips. 1693, p. 252, Art. 3. 16. A lamina, in the form of a complete cycloid, moves through a fluid in the direction of its axis, first with its vertex and next with its base foremost: to compare the resistances in the two cases. The resistance in the former case will be to that in the latter as 3 to 4. Herman: Phoronomia, p. 249. 17. To compare the resistance on a sphere placed in a stream with that on a circular plate of the same radius placed at right angles to the stream. Resistance on the sphere = half that on the plate. Newton Principia, lib. 11. Prop. 34. Bossut: Traité d' Hydrodynamique, tom. I. p. 439. 18. A solid segment of a sphere moves through a fluid in the direction of its axis, first with its vertex and next with its base foremost to compare the resistances in the two cases. If R, R', be the resistances in the former and latter case respectively, r the radius of the sphere and y the radius of the base of the segment, R: Ry: r2. Herman: Phoronomia, p. 248. 19. A segment of a paraboloid of revolution moves through. a fluid in the direction of its axis, to which its base is perpendicular, first with its vertex and next with its base foremost: to compare the resistances in the two cases. If y denote the diameter of the base, and the latus rectum of the generating parabola, the resistance in the former case will be to that in the latter in a ratio denoted by the expression y log (1+). Herman: Phoronomia, p. 248. 20. To determine the resultant resistance on a spheroid moving through a fluid in the direction of its axis. If 2a, 2b, be the axes of the generating ellipse, of which the former is the axis of revolution, and v be the velocity of the spheroid, the required resistance will be equal to 21. To determine the velocity of the wind, blowing horizontally, when it is just able to overturn a given circular cylinder standing on a horizontal plane and prevented from sliding. If v represent the velocity and p the density of the wind, W the weight and 7 the length of the cylinder, 22. A semicircular arc, of which the radius is a, vibrates about an axis through one extremity and perpendicular to its plane, which is vertical, in a medium where the direct resistance if v were on an element ds of the arc would be equal to λp gvds k the normal velocity of ds, λ denoting the area of a transverse section and p the density of the arc: to find the length of the isochronous simple pendulum. The length of the isochronous simple pendulum is equal to CHAPTER IV. MOTION OF SOLID BODIES ACTED ON BY FLUIDS. SECTION I. Finite Vertical Oscillations in Incompressible Fluids. IN the solutions of the class of problems which form the subject of this section we shall confine our attention to the motion of floating bodies as subject to the mere statical pressure of the circumambient fluid, without taking account of the modification of the effect of such pressure which arises from the resistance and inertia of the fluid. It will easily be imagined that, such an hypothesis being adopted, the investigation of the motion of floating bodies will be of little value except as affording analytical exercise, and that the results obtained will be entirely at variance with the experiment. The premises which form the basis of this section are chosen merely to simplify the algebraical formula which on any true hypothesis would ordinarily be so complicated as to present insuperable difficulties to the analyst. 1. A solid cylinder of given length is pressed down in a vertical position into a fluid, so that its upper end is on a level with the surface, the specific gravity of the cylinder being onehalf that of the fluid: the pressure being removed, to find the greatest height to which the upper end of the cylinder will rise above the surface of the fluid. Let a denote the height of the upper end of the cylinder above the surface of the fluid after any time t from the commencement of the motion; 2a the length of the cylinder; the area of its base, p the density of the cylinder. Then, for the A and B being arbitrary constants. But, initially, x = 0 and 2. A solid cylinder, resting in a fluid which is contained in a hollow cylinder with the same vertical axis, is thrust vertically downwards; to ascertain the period of an oscillation. Let r, r', be the radii of the solid and of the hollow cylinder respectively; p the density of the solid cylinder and p' of the fluid. Also, let a represent the altitude of the base of the solid cylinder and x' of the surface of the fluid, above the base of the hollow cylinder, at any time t. Then, for the motion of the oscillating cylinder, there is But, c denoting the volume of the fluid, π2x' - πr2 (x' - x) = c3, 3. A solid, the surface of which is generated by the revolu tion of the curve n .1 2 ухх round the axis of x, floats in a fluid with a portion h of its axis immersed to find how much the solid must be depressed, that it may, on its return, just emerge from the fluid. The required depth of depression is equal to 1) h. 4. A sphere, of specific gravity, p is immersed in a fluid of specific gravity p', the depth of the centre of the sphere below the surface of the fluid being equal to n times its radius: supposing the sphere to be allowed to ascend, to find the values of n, first, that it may rise just half out of the fluid, and, secondly, that it may rise just entirely out. The two required values of n are |