A Treatise on the Theory of Screws

316 THE THEORY OF SCREWS. [296- from it by <p. In like manner if <£ be an impulsive screw corresponding to p, as instantaneous screw we have another locus parallel to p for the centre of gravity. But as the centre of gravity is at infinity these two loci must there intersect, i.e. they must be parallel and so must X and p, and hence all instantaneous screws must be parallel. Thus we see that all the screws on A must be parallel, i.e. that A must have degraded into an extreme type of cylindroid. 297. Another extreme Case. Given any two cylindroids A and P it is, as we have seen, generally possible to correlate in one way the several screws on A to those on P so that an impulsive wrench given to a certain rigid body about any screw on P would make that body commence to move by twisting about its cor- respondent on A. One case of failure has just been discussed. The case now to be considered is not indeed one of failure but one in which a,ny two pairs of screws on A and P will stand to each other in the desired relations. Suppose that A and P happened to fulfil the single condition that each of them shall contain one screw which is reciprocal to the other cylindroid. We have called the cylindroids so circumstanced “ normal.” Let X be the screw on A which is reciprocal to every screw on P. If then P and A are to stand to each other in the required relation, X must be reciprocal to its impulsive screw. But this is only possible on one condition. The mass of the body must be zero. In that case, if there is no mass involved any one of the screws on P may be the impulsive screw corresponding to any one of the screws on A. Here again the question arises as to what becomes of the homographic equation which defines so precisely the screw on P which corresponds to the screws on A (§ 295). It might have been expected that in the case of two normal cylindroids this homographic equation should become evan- escent. But it does not do so. But there is no real contradiction. The greater includes the less. If every screw on P will suit as correspondent every screw on A then å fortiori will the pairs indicated by the homography fulfil the conditions requisite. 1’hat any two pairs of screws will be correspondents in this case is obvious from the following.