A Treatise on the Theory of Screws

196] FREEDOM OF THE THIRD ORDER. 193 195. Wrench evoked by Displacement. By the aid of the quadric of the potential we shall be able to solve the problem of the determination of the screw on which a wrench is evoked by a twist about a screw 0 from a position of stable equilibrium. The construction which will now be given will enable us to determine the screw of the system on which the reduced wrench acts. Draw through the centre of the pitch quadric a line parallel to 0. Con- struct the diametral plane A of the quadric of the potential conjugate to this line, and let X, fj. be any two screws of the system parallel to a pair of conjugate diameters of the quadric of the potential which lie in the plane A. Then the required screw </> is parallel to that diameter of the pitch quadric which is conjugate to the plane A. For (fi will then be reciprocal to both X and /z.; and as X, /z, 0 are conjugate screws of the potential, it follows that a twist about 0 must evoke a reduced wrench on <f>. 196. Harmonic Screws. When a rigid body has freedom of the third order, it must have (§ 106) three harmonic screws, or screws which are conjugate screws of inertia, as well as conjugate screws of the potential. We are now enabled to construct these screws with facility, for they must be those screws of the screw system which are parallel to the triad of conjugate diameters common to the ellipsoid of inertia, and the quadric of the potential. We have thus a complete geometrical conception of the small oscillations of a rigid body which has freedom of the third order. If the body be once set twisting about one of the harmonic screws, it will continue to twist thereon for ever, and in general its motion will be compounded of twisting motions upon the three harmonic screws. If the displacement of the body from its position of equilibrium has been effected by a small twist about a screw on the cylindroid which contains two of the harmonic screws, then the twist can be decomposed into com- ponents on the harmonic screws, and the instantaneous screw about which the body is twisting at any epoch will oscillate backwards and forwards upon the cylindroid, from which it will never depart. If the periods of the twist oscillations on two of the harmonic screws coincided, then every screw on the cylindroid which contains those harmonic screws would also be a harmonic screw. If the periods of the three harmonic screws were equal, then every screw of the system would be a harmonic screw. B. 13