A Treatise on the Theory of Screws

344 THE THEORY OF SCREWS. [317, Draw pairs of corresponding planes through X and X'. The locus of their intersection will be a quadric S", which also contains the four double points. S' and C, being of the second and the third order respectively, will intersect in six points. Two of these are on X and X', and are thus dis- tinguished. The four remaining intersections will be the required double points, and thus the problem has been solved. These double points correspond to the principal screws of inertia, which are accordingly determined. In the case of freedom of the fifth order, the geometrical analogies which have hitherto sufficed are not available. We have to fall back on the general fact that the impulsive screws and the corresponding instantaneous screws form two homographic systems. There are five double screws be- longing to this hornography. These are the principal screws of inertia. 318. Correlation of Two Systems of the Third Order. It being given that a certain system of screws of the third order, P, is the locus of impulsive screws corresponding to another given system of the third order, A, as instantaneous screws, it is required to correlate the corre- sponding pairs on the two systems. We have already had frequent occasion to use the result demonstrated in § 293, namely, that when two impulsive and instantaneous cylindroids were known, we could arrange the several screws in corresponding pairs without any further information as to the rigid body. We have now to demonstrate that when we are given an impulsive system of the third order, and the corresponding instantaneous system, there can also be a similar adjustment of the corresponding pairs. It has first to be shown, that the proposed problem is a definite one. The data before us are sufficient to discriminate the several pairs of screws, that is to say, the data are sufficient to point out in one system the corre- spondent to any specified screw in the other system. We have also to show that there is no ambiguity in the solution. There is only one rigid body (§ 293) which will comply with the condition, and it is not possible that there could be more than one arrangement of corresponding pairs. Let a, S, 7 be three instantaneous screws from A, and let their corre- sponding impulsive screws be y, %, %, in P. In the choice of a screw from a system of the third order there are two disposable quantities, so that, in the selection of three correspondents in P, to three given screws in A, there would be, in general, six disposable coordinates. But the fact that a, r) and ß, £ are two pairs of correspondents necessitates, as we know,