A Treatise on the Theory of Screws

342 THE THEORY OF SCREWS. [316, The simplest method of solving this problem is that already given in § 139, in which we regard the six points, suitably arranged, as the vertices of a hexagon; then the Pascal line of the hexagon intersects the circle in two points which are the points corresponding to the principal screws of inertia. 317. Third and Higher Systems. We next investigate the principal screws of inertia of a body which has three degrees of freedom. We have first, by the principles already ex- plained, to discover four pairs of correspondents. When four such pairs aie known, the principal screws of inertia can be constructed. Perhaps the best method of doing so is to utilize the plane correspondence, as explained in Chap. xv. The corresponding systems of impulsive screws and instantaneous screws, in the system of the third order, are then repre- sented by the homographic systems of points in the plane. When four pairs of such correspondents are known, we can construct as many additional pairs as may be desired. Let a, ß, y, <5 be four points in the plane, and let v, %, £ 0 be the points corresponding, so that represents the impulsive screw, and a the instantaneous screw, and similarly for the other pairs. Let it be required tu find the impulsive screw </>, which corresponds to any fifth instantaneous sciew e. Since anharmonic ratios are the same m two corresponding figures, we have a(ß, y, 8, e)=v(£, 0, <£), thus we get one ray which contains </>. We have also ß («, 7, 8, e) = £ (v, £ 0, which gives a second ray £</>, containing </>, and thus </> is known. A construction for the double points of two homographic systems of points in the same plane is as follows:— Let 0 and O' be a pair of corresponding points. Then each ray through 0 will have, as its correspondent, a ray through O'. The locus of the intersection of these rays will be a conic S. This conic S must pass through the three double points, and also through 0 and O'. Diaw the conic »S' , which is the locus of the points in the second system corresponding to the points on S, regarded as in the first system. Then since 0 lies on S, we must have 0' on S'. But S' must also pass through the three double points. O’ is one of the four intersections of Ä and S', and the three others are the sought double points. Thus the double points are constructed. Aud in this manner we obtain the three principal screws of inertia in the case of the system of the third order.