A Treatise on the Theory of Screws

320] the geometrical theory. 349 radius of gyration about the remaining conjugate diameter. The data before us are not adequate to the removal of this indefiniteness. It must be remembered that the data in such a case are just so many but no more than suffice for the specification of the n-system A. The number of data neces- sary to define an n-system is n (6 — n). If, as in the present case, n = 4, the number of data is 8. We are thus one short in the number of data necessary to specify a rigid body. Thus we confirm the result previously obtained. We can assert that for any one of the singly infinite number of rigid bodies which fulfil the necessary conditions, the system A will be the locus of the instantaneous screws which correspond to the screws of the system P as impulsive screws. Though in the two cylindroids A' and P' we are able to establish the several pairs of correspondents quite definitely, yet we must not expect, with the data before us, to be able to correlate the pairs of screws in A and P definitely. If this could be done then the rigid body would be quite deter- minate, which we know is not the case. There is, however, only a single indeterminate element left in the con-elation of the screws in A with the screws of P. This we prove as follows:— Let if) be any screw of P on which an impulsive wrench is to act. Let 8 be the instantaneous screw in A about which the movement commences. We shall now show that though 8 cannot be completely defined, in the absence of any further data, yet it can be shown to be limited to a certain cylindroid. Let G be the centre of gravity. Then we know that an impulsive force directed through G will generate a movement of translation in a direction parallel to the force. Such a movement may, of course, be regarded as a twist about a screw of infinite pitch. Draw through G a plane normal to </>. Any screws of infinite pitch in this plane will be reciprocal to </>. It follows from the laws of conjugate screws of inertia that the impulsive forces in this plane, by which translations could be produced, must lie on screws of zero pitch which are reciprocal to 8. Take any two of such screws: then we know that S is reciprocal to these two screws and also to P'. It follows that S is reciprocal to the screws of a determinate system of the fourth order, and therefore 8 must lie on a deter- mined cylindroid. We may commence to establish the correspondence between P and A by choosing some arbitrary screw </> on P, and then drawing the cylindroid on A, on which we know that the instantaneous screw corresponding to P must lie. Any screw on this cylindroid may be selected as the instantaneous screw which corresponds to </>. Once that screw 8 had been so chosen there