A Treatise on the Theory of Screws

332 THE THEORY OF SCREWS. [307- Let the centre of gravity be constructed as in the last section; then an impulsive force through the centre of gravity will produce the velocity of translation on Let us denote by the screw of zero pitch on which this force lies. We thus have as the impulsive screw corresponding to the instan- taneous screw S15 while is the impulsive screw corresponding to the instantaneous screw <50. Draw "now the cylindroids (^, -i/r) and (81; 80). The first of these is the locus of the impulsive screws corresponding to the instantaneous screws on the second. As already explained, we can completely correlate the screws on two such cylindroids. We can, therefore, construct the impulsive screw on (y, yfr) which corresponds to any instantaneous screw on (£>1; 80). It is, however, obvious, from the construction, that the original screw 3 lies on the cylindroid (81; S„). Hence we obtain the impulsive screw which corresponds to ö as the instantaneous screw, and the problem has been solved. 308. Another method. We might have proceeded otherwise as follows:—From the three given pairs of impulsive screws and instantaneous screws ya, %ß, £y we can find other pairs in various ways. For example, draw the cylindroids (a, ß) and (£, £); then select, by principles already explained, a screw 8 on the first cylindroid, and its correspondent 3 on the second. In like manner, from the cylindroids (a, 7) and (77, £), we can obtain another pair (</>, e). We have thus five pairs of correspondents, ya, Irß, £7, 38, </>e. Each of these will give a diameter of the momental ellipsoid, and the radius of gyration about that diameter. Thus we know the centre of the momental ellipsoid and five points on its surface. The ellipsoid can be drawn accordingly. Its three principal axes give the principal screws of inertia. All other pairs of correspondents can then be determined by a construction given later on (§311). 309. Unconstrained motion in system of second order. Suppose that a cylindroid be drawn through any two (not lying along the same principal axis) of the six principal screws of inertia of a free rigid body. If the body while at rest be struck by an impulsive wrench about any one of the screws of the cylindroid it will commence to move by twisting about a screw which also lies on the cylindroid. For the given impulsive wrench can be replaced by two component wrenches on any two screws of the cylindroid. We shall, accordingly, take the component wrenches of the given impulse on the two principal screws of inertia which, by hypothesis, are contained on the cylindroid. Each of those components