A Treatise on the Theory of Screws

308 THE THEORY OF SCREWS. [290- about some corresponding screw on A, and the two systems of screws would have chiastic homography. If the body were given both in constitution and in position, then, of course, there would be nothing arbitrary in the choice of the corresponding screws. Suppose, however, that a screw rj had been chosen arbitrarily on B to correspond to a screw a on 4, it would then be generally possible to design and place a rigid body so that it should begin to twist about a in consequence of the impulse on y. There would, however, be no arbitrary element remaining in the homography. Thus, we see that, while for homography, in general, three pairs of correspondents can be arbitrarily assigned, there can only be two pairs so assigned for chiastic homography, while for such a particular type as that which relates to impulsive screws and the corresponding instantaneous screws, only one pair can be arbitrarily chosen. 291. Case of Normal Cylindroids. We have already had occasion (§ 118), to remark on the curious relation- ship of two cylindroids when a screw can be found on either cylindroid which is reciprocal to all the screws on the other. If, for the moment, we speak of two such cylindroids as “ normal,” then we have the following theorem:— Any homography of the screws on two cylindroids must be chiastic if the two cylindroids are normal. Let a, ß, 7 be any three screws on one cylindroid, and y, %, £ any three screws on the other; then, since the cyliudroids are normal, we have ^ßi ^yn > ^a^ß( = whence we obtain unless therefore £ is reciprocal to ß, we must have If, however, £ had been reciprocal to ß, then one of these screws (suppose ß) must have been the screw on its cylindroid reciprocal to the entire group of screws on the other cylindroid. In this case we must have ®rß7;=:0; so that even in this case it would still remain true that ^a^ß^yf — ^a^ß^ya = 0. It is, indeed, a noteworthy circumstance that, for any and every three pairs of screws on two normal cylindroids, the relation just written must be fulfilled.