A Treatise on the Theory of Screws

195 FREEDOM OF THE THIRD ORDER. 197] diameters common to the two ellipses in which the plane conjugate to the vertical axis in the momenta! ellipsoid cuts the momental ellipsoid and the cylinder. These three lines are the three harmonic axes. As to that vertical axis which appears to be one of the harmonic axes, the time of vibration about it would be infinite. The three har- monic screws which are usually found in the small oscillations of a body with freedom of the third order are therefore reduced in the present case to two, and we have the following theorem :— A rigid body which is free to rotate about a fixed point is at rest under the action of gravity. If a plane S be drawn through the point of suspension 0, conjugate to the vertical diameter 01 of the momental ellipsoid, then the common conjugate diameters of the two ellipses in which S cuts the momental ellipsoid, and a circular cylinder whose axis is 01, are the two harmonic axes. If the body be displaced by a small rotation about one of these axes, the body will continue for ever to oscillate to and fro upon this axis, just as if the body had been actually constrained to move about this axis. To complete the solution for any initial circumstances of the rigid body, a few additional remarks are necessary. Assuming the body in any given position of equilibrium, it is first to be displaced by a small rotation about an axis OX. Draw the plane containing 01 and OX, and let it cut the plane S in the line OY. The small rotation around OX may be produced by a small rotation about 01, followed by a small rotation about OY. The effect of the small rotation about 01 is merely to alter the azimuth of the position, but not to disturb the equi- librium. Had we chosen this altered position as that position of equilibrium from which we started, the initial displacement would be communicated by a rotation around OY. We may, therefore, without any sacrifice of generality, assume that the axis about which the initial displacement is imparted lies in the plane We shall now suppose the body to receive a small angular velocity about any other axis. This axis must be in the plane S, if small oscillations are to exist at all, for the initial angular velocity, if not capable of being resolved into components about the two harmonic axes, will have a component around the vertical axis 01. An initial rotation about 01 would give the body a continuous rotation around the vertical axis, which is not admissible when small oscillations only are considered. If, therefore, the body performs small oscillations only, we may regard the initial axis of displacement as lying in the plane S, while we must have the initial instantaneous axis in that plane. The initial displacement may be resolved into two displacements, one on each of the harmonic axes, and 13—2