A Treatise on the Theory of Screws

348 THE THEORY OF SCREWS. [319, Let P' be the cylindroid which is composed of the screws reciprocal to P. Let Plt P2, P-i, P4 be any four impulsive screws on P. Let j41; A.,,, A3, At be the four corresponding instantaneous screws on A. Take any screw a on the cylindroid P'. Let // be the corresponding impulsive screw. Since a is reciprocal to all the screws on P it must be reciprocal to Pj. It follows from the fundamental property of conjugate screws of inertia, that y must be reciprocal to A3. In like manner we can show that y is reciprocal to A.,, A:i, and A4. It follows that y is reciprocal to the whole system A, and therefore must be contained in the reciprocal cylindroid A'. Hence we obtain the following remarkable result, which is obviously generally true, though our proof has been enunciated for the system of the fourth order only. Let P and A be any two systems of screws of the nth order, and P' and A' their respective reciprocal systems of the (f~n)th order. If P be the collec- tion of impulsive screws corresponding severally to the screws of A as the instantaneous screws for a certain free rigid body; then, for the same free rigid body A' will be the collection of impulsive screws which correspond to the screws of P' as instantaneous screws. 320. Systems of the Fourth Order. Thus we see that when we are given two systems of the fourth order P and A as correspondingly impulsive and instantaneous, we can immediately infer that, for the same rigid body, the screws on the cylindroid A' are the impulsive screws corresponding to the instantaneous screws on the cylindroid P'. We can now make use of that instructive theorem (§ 293) which declares that when two given cylindroids are known to stand to each other in this peculiar relation, we are then able, without any further information, to mark out on the cylindroids the corresponding pairs of screws. We can then determine the centre of gravity of the rigid body on which the impulsive wrenches act. We can find a triad of conjugate diameters of the momental ellipsoid, and the radii of gyration about two of those diameters. Hence we have the following result:— If it be given that a certain system of the fourth order is the locus of the impulsive screws corresponding to the instantaneous screws on another given system of the fourth order, the body being quite unconstrained, we can then determine the centre of gravity of the body, we can draw a triad of the conjugate diameters of its momental ellipsoid, and we can find the radii of gyration about two of those diameters. There is still one undetermined element in the rigid body, namely, the