A Treatise on the Theory of Screws

319] THE GEOMETRICAL THEORY. 347 rigid body such that, if that body, while at rest and unconstrained, receive an impulsive wrench about any screw of the first system, the instantaneous move- ment will be a twist about a screw of the second system. The two systems of corresponding impulsive and instantaneous screws on the two systems of the third order, form two homographic systems. There are, of course, infinite varieties in the possible homographic cor- respondences between the screws of two systems of the third order. The number of such correspondences is just so many as the number of possible homographic correspondences of points in two planes. There is, however, only one correspondence which will fulfil the peculiar requirements when one of the systems expresses the instantaneous screws, and the other the impulsive screws severally corresponding thereto. If we are given one pair of corresponding impulsive and instantaneous screws, the body is not by such data fully determined. We are only given five coordinates, and four more remain, therefore, unknown. If we are given two con-esponding impulsive cylindroids and instantaneous cylindroids, the body is still not completely specified. We have seen how eight of its coordinates are determined, but there is still one remaining indeterminate. If we are given a system of the fourth order of impulsive screws, and the corresponding system of the fourth order of instantaneous screws, the body, as in the other cases, remaining perfectly free, there are also, as we shall see in the next section, a singly infinite number of rigid bodies which fulfil the necessary conditions. In like manner, it will appear that, if we are given a system of the fifth order consisting of impulsive screws, and a corresponding system of the fifth order consisting of instantaneous screws, the body has really as much indeterminateness as if we had only been given a single pair of corresponding screws. But the case of two systems of the third order is exceptional, in that when it is known that one of these is the locus of the instantaneous screws, which correspond to the screws of the other system regarded as impulsive screws, the rigid body for which this state of things is possible is completely and uniquely specified as to each and every one of its nine coordinates. 319. A Property of Reciprocal Screw Systems. Given a system of the fourth order A and another system of the fourth order P. If it be known that the latter is the locus of the screws on which must lie the impulsive wrenches which would, if applied to a free rigid body, cause instantaneous twist velocities about the several screws on A, let us consider what can be inferred as to the rigid body from this fact alone. Let A' be the cylindroid which is composed of the screws reciprocal to A.