A Treatise on the Theory of Screws

352 THE THEORY OF SCREWS. [323 follows, that a wrench on the screw of zero pitch, which lies on the ray Gj G2, will have the same instantaneous screw whether that wrench be applied to M or to M We have now to examine whether there can be any other pair of im- pulsive and instantaneous screws in the same circumstances. Let us suppose that when 6 assumes a certain position y, we have X and p coalescing into the single screw a. We know that the centre of gravity lies in a plane through a, and the shortest distance between a and We know, also, that da =pa tan (a?;), where is the distance of the centre of gravity from a. It therefore follows that a must be parallel to Gfl.,. We have, however, already had occasion (§ 303) to prove that, if be the radius of gyration of a body about a ray through its centre of gravity, parallel to a, Hence it appears that, for the required condition to be satisfied, each of the two bodies must have the same radius of gyration about the axis through its centre of gravity, which is parallel to a. Of course this will not generally be the case. It follows that, in general, there cannot be any such pair of impulsive screws and instantaneous screws, as has been supposed. Hence we have the following result:— Two rigid bodies, with different centres of gravity, have, in general, no other common pair of impulsive screws and instantaneous screws than the screw, of zero pitch, on the ray joining the centres of gravity, and the screw of infinite pitch parallel thereto. We shall now consider what happens when the exceptional condition, just referred to, is fulfilled, that is, when the radius of gyration of the ray G\G., is the same for each of the bodies. In each of the momental ellipsoids about the centres of gravity of the two bodies, draw the plane conjugate to the ray Gfff. Let these planes intersect in a ray T. Suppose that an impulsive force, directed along T, be made to act on the body whose centre of gravity is Gx. It is plain, from Poinsot’s well-known theory, that the rotation produced by such an impulse will be about a ray parallel to Gff. If this impulsive wrench had been applied to the body whose centre of gravity is Ga, the instantaneous screw would also be parallel to GAG2. If we now introduce the condition that the radius of gyration of each of the bodies, about G1G2, is the same, it can be easily deduced that the two instantaneous screws are identical. Hence we see that T, regarded as an impulsive screw of zero pitch, will have the same instantaneous screw for each of the two bodies.