A Treatise on the Theory of Screws

326 THE THEORY OF SCREWS. [305, Let G be this intersection, and draw GP parallel to a and equal to the radius of gyration about GP, which we have shown to be known from the fact that a and r) are known. Let X be the plane conjugate to GP in the momenta! ellipsoid, then this plane is also known. In like manner, draw GQ parallel to ß and equal to the radius of gyration about GQ. Let Y be the plane, conjugate to GQ, in the momental ellipsoid. Let Pj and P2 be the perpendiculars from P, upon X and Y respec- tively. Let Q, and Q2 be the perpendiculars from Q, upon X and Y respec- tively. Then, from the properties of the ellipsoid, it is easily shown that P^.P^Q^Q.. This is the second geometrical relation between the two pairs of screws a, 7) and ß, %. Subject to these two geometrical conditions or to the two formulae to which they are equivalent the two pairs of screws might be chosen arbitrarily. As these two relations exist, it is evident that the knowledge of a second pair of corresponding impulsive screws and instantaneous screws cannot bring five independent data as did the first pair. The second pair can bring no more than three. From our knowledge of the two pairs of screws together we thus obtain no more than eight data. We are consequently short by one of the number requisite for the complete specification of the rigid body in its abstract form. It follows that there must be a singly infinite number of rigid bodies, every one of which will fulfil the necessary conditions with reference to the two pairs of screws. For every one of those bodies a. is the instantaneous screw about which twisting motion would be produced by an impulsive wrench on i). For every one of those bodies ß is the instantaneous screw about which twisting motion would be produced by an impulsive wrench on 306. A System of Rigid Bodies. We have now to study the geometrical relations of the particular system of rigid bodies in singly infinite variety which stand to the four screws in the relation just specified. Draw the cylindroid (a, ß) which passes through the two screws a and ß. Draw also the cylindroid (77, 0 which passes through the two corresponding impulsive screws 97 and It is easily seen that every screw on the first of these cylindroids if regarded as an instantaneous screw, with respect to the same rigid body, will have its corresponding impulsive screw on the second