A Treatise on the Theory of Screws

76] THE EQUILIBRIUM OF A RIGID BODY. 65 constraints are only manifested by the success with which they resist the efforts of certain wrenches to disturb the equilibrium of the body. 75. Parameters of a Screw System. We next consider the number of parameters required to specify a screw system of the nth order often called for brevity an ».-system. Since the system is defined when n screws are given, and since five data are required for each screw, it might be thought that bn parameters would be necessary. It must be observed, however, that the given bn data suffice not only for the purpose of defining the screw system but also for pointing out n special screws upon the screw system, and as the pointing out of each screw on the system requires n — 1 quantities (§ 69), it follows that the number of parameters actually required to define the system is only bn — n (n — 1) = n (6 — n). This result has a very significant meaning in connexion with the theory of reciprocal screw systems P and Q. Assuming that the order of P is n, the order of Q is 6 — n; but the expression w (6 — n) is unaltered by changing n into 6 — n. It follows that the number of parameters necessary to specify a screw system is identical with the number necessary to specify its reciprocal screw system. This remark is chiefly of importance in connexion with the systems of the fourth and fifth orders, which are respectively the reciprocal systems of a cylindi’oid and a single screw. We are now assured that a collection of all the screws which are reciprocal to an arbitrary cylindroid can be nothing less than a screw system of the fourth order in its most general type, and also, that all the screws in space which are reciprocal to a single screw must form the most general type of a screw system of the fifth order. 76. Applications of Co-ordinates. If the co-ordinates of a screw satisfy n linear equations, the screw must belong to a screw system of the order 6 — n. Let r) be the screw, and let one of the equations be + ... + Ar,ye=O, whence y must be reciprocal to the screw whose co-ordinates are pro- portional to A1~, ... Aa, (§ 37). Pi p<* ’ It follows that t? must be reciprocal to n screws, and therefore belong to a screw system of order 6 — n. Let a, ß, y, 8 be for example four screws about which a body receives twists of amplitudes a, ß', y', S'. It is required to determine the screw p and