A Treatise on the Theory of Screws

194 THE THEORY OF SCREWS. f 197 197. Oscillations of a Rigid Body about a Fixed Point*. We shall conclude the present Chapter by applying the principles which it contains to the development of a geometrical solution of the following important problem:— A rigid body, free to rotate in every direction around a fixed point, is in stable equilibrium under the influence of gravity. The body is slightly disturbed: it is required to determine its small oscillations. Since three co-ordinates are required to specify the position of a body when rotating about a point, it follows that the body has freedom of the thud Older. Pho scrow system, however, assumes a very extreme type, for the pitch quadric has become illusory, and the screw system reduces to a pencil of screws of zero pitch radiating in all directions from the fixed point. The quantity ue appropriate to a screw 9 reduces to the radius of gyration when the pitch of the screw is zero; hence the ellipsoid of inertia reduces in the present case to the well-known momental ellipsoid. The quadric of the potential (§ 193) assumes a remarkable form in the present case. The work done in giving the body a small twist is propor- tional to the vertical distance through which the centre of inertia is elevated. In the position of equilibrium the centre of inertia is vertically beneath the point of suspension, it is therefore obvious from symmetry that the ellipsoid of the potential must be a surface of revolution about a vertical axis. It is further evident that the vertical radius vector of the cylinder must be infinite, because no work is done in rotating the body around a vertical axis. Let 0 be the centre of suspension, and I the centre of inertia, and let OP be a radius vector of the quadric of the potential. Let fall IQ per- pendicular on OP, and PT perpendicular upon 01. It is extremely easy to show that the vertical height through which I is raised is proportional to ZQ2x OF2; whence the area of the triangle OPT is constant, and there- fore the locus of P must be a right circular cylinder of which 01 is the axis. We have now to find the triad of conjugate diameters common to the momental ellipsoid, and the circular cylinder just described. A group of three conjugate diameters of the cylinder must consist of the vertical axis, and any two other lines through the origin, which are conjugate diameters of the ellipse in which their plane cuts the cylinder. It follows that the triad required will consist of the vertical axis, and of the pair of conjugate * Trims, Roy. Irish Acad., Vol. xxiv. Science, p. 593 (1870).