A Treatise on the Theory of Screws

250 THE THEORY OF SCREWS. [230- It must always be possible to find a single screw X which is reciprocal to the six screws P, Q, R, S, T, U. Suppose the rigid body were only free to twist about X, then the six forces would not only collectively be in equi- librium, but severally would be unable to stir the body only free to twist about X. In general a body able to twist about six screws (of any pitch) would have perfect freedom; but the body capable of rotating about each of the six lines, P, Q, R, S, T, U, which are in involution, is not necessarily perfectly free (Mobius). If a rigid body were perfectly free, then a wrench about any screw could move the body; if the body be only free to rotate about the six lines in involution, then a wrench about every screw (except X) can move it. The conjugate axes discussed by Sylvester are presented in the Theory of Screws as follows:—Draw any cylindroid which contains the reciprocal screw X, then the two screws of zero pitch on this cylindroid are a pair of conjugate axes. For a force on any transversal intersecting this pair of screws is reciprocal to the cylindroid, and is therefore in involution with the original system. Draw any two cylindroids, each containing the reciprocal screw, then all the screws of the cylindroids form a screw system of the third order. Therefore the two pairs of conjugate axes, being four screws of zero pitch, must lie upon the same quadric. This theorem, due to Sylvester, is proved by him in a different manner. The cylindroid also presents in a clear manner the solution of the problem of finding two rotations which shall bring a body from one position to any other given position. Find the twist which would effect the desired change. Draw any cylindroid through the corresponding screw, then the two screws of zero pitch on the cylindroid are a pair of axes that fulfil the required conditions. If one of these axes were given the cylindroid would be defined and the other axis would be determinate. 231. Four Screws of a Five-system on every Quadric. On any single sheeted hyperboloid four screws of any given pitch p can in general be determined which belong to any given system of the fifth order. A pair of these screws lie on each kind of generator. Let X be the screw reciprocal to the system. Take any three generators A, B, C of one system on the hyperboloid, and regarding them as screws of pitch p draw the cylindroid XA and tak© on this A' th6 second screw of pitch -p. Then the two screws of pitch p which can be drawn as transversals across A, B, C, A' are coincident with two generators of the hyperboloid