A Treatise on the Theory of Screws

186] FREEDOM OF THE THIRD ORDER. 187 Any line in space when it receives the proper pitch is a screw of this system. Through any point in space a plane can be drawn such that every line in the plane passing through the point with zero pitch is a screw of the system (§ 110). Finally, if the body has only freedom of the third order, the four equi- librating forces P, Q, R, 8 may be situated anywhere. The positions of the forces being given, their magnitudes are determined; for draw three screws Xlt Xs, X3 reciprocal to the system, and find (§ 28) the intensities of the seven equilibrating wrenches on P, Q, R, 8, Xlt X3, X3. The last three are neutralised by the reactions of the constraints, and the four former must therefore equilibrate. Given any four screws in space, it is possible for four wrenches of proper intensities on these screws to hold a body having freedom of the third order in equilibrium. For, take the four given screws, and three reciprocal screws. Wrenches of proper intensities on these seven screws will equilibrate; but those on the reciprocal screws are destroyed by the reactions, and, therefore, the four wrenches on the four screws equilibrate. It is manifest that this theorem may be generalised into the following:—If a body have freedom of the &th order, then properly selected wrenches about any Æ + 1 screws (not reciprocal to the screw system) will hold the body in equilibrium. That a rigid body with freedom of the third order may be in equilibrium under the action of gravity, we have the necessary and sufficient condition, which is thus stated :— The vertical through the centre of inertia must be one of the reciprocal system of generators on the pitch quadric. We see that the centre of inertia must, therefore, lie upon a screw of zero pitch which belongs to the screw system; whence we have the following theorem:—The restraints which are necessary for the equilibrium of a body which has freedom of the third order under the action of gravity, would permit rotation of the body round one definite line through the centre of inertia. 186. The Ellipsoid of Inertia. The momental ellipsoid, which is of such significance in the theory of the rotation of a rigid body about a fixed point, is presented in the Theory of Screws as a particular case of another ellipsoid, called the ellipsoid of inertia, which is of great importance in connexion with the general screw system of the third order.