A Treatise on the Theory of Screws

323] THE GEOMETRICAL THEORY. 353 If we regard T and G^G^ as two screws of zero pitch, and draw the cylindroid through these two screws, then any impulsive wrench about a screw on this cylindroid will have the same instantaneous screw for either of the two bodies to which it is applied. For such a wrench may be decomposed into forces on T and on G!16r2; these will produce, in either body, a twist about a, and a translation parallel to a, respectively. We therefore obtain the following theorem:— If two rigid bodies have different centres of gravity, Cq and G2, and if their radii of gyration about the ray GAG2 are equal, there is then a cylindroid of screws such that an impulsive wrench on any one of these screws will make either of the rigid bodies begin to twist about the same screw, and the instantaneous screws which correspond to the several screws on this cylindroid, all lie on the same ray G^G^, but with infinitely varied pitch. It is to be remarked that under no other circumstances can any im- pulsive screw, except the ray Gjöa, with zero pitch, have the same instan- taneous screw for each of the two bodies, so long as their centres of gravity are distinct. We might have demonstrated the theorem, above given, from the results of § 303. We have there shown that, when an impulsive screw and the corresponding instantaneous screw are given, the rigid body must fulfil five conditions, the nature of which is fully explained. If we take two bodies which comply with these conditions, it appears that the ray through their centre of gravity is parallel to the instantaneous screw, and we also find that their radii of gyration must be equal about the straight line through their centres of gravity. If two rigid bodies have the same centre of gravity, then, of course, any ray through this point will be the seat of an impulsive wrench on a screw of zero pitch such that it generates a twist velocity on a screw of infinite pitch, parallel to the impulsive screw. This will be the case to whichever of the two bodies the force be applied. We have therefore a system of the third order (much specialized no doubt) of impulsive screws, each of which has the same instantaneous screw for each of the two bodies. In general there will be no other pairs of common impulsive and instantaneous screws beyond those indicated. Under certain circumstances, however, there will be other screws possess- ing the same relation. We may suppose the two momental ellipsoids to be drawn about the common centre of gravity. These ellipsoids will, by a well-known property, possess one triad of common conjugate diameters. In general, of course, the b. 23