A Treatise on the Theory of Screws

30 THE THEORY OF SCREWS. [25-27 25 Screw Reciprocal to Five Screws. The determination of a screw reciprocal to five given screws must in general admit of only a finite number of solutions, because the number of conditions to be fulfilled is the same as the number of disposable constants. It is very important to observe that this number must be unity. For if two screws could be found which fulfilled the necessary conditions, then these conditions would be equally fulfilled by every screw on the cylindroid determined by those screws (§ 22), and therefore the number of solutions of the problem would not be finite. The construction of the screw whose existence is thus demonstrated, can be effected by the results of the last article. Take any four of the five screws, and draw the reciprocal cylindroid which must contain the required screw. Any other set of four will give a different cylindroid, which also contains the required screw. These cylindroids must therefore intersect in the single screw, which is reciprocal to the five given screws. 26. Screw upon a Cylindroid Reciprocal to a Given Screw. Let e be the given screw, and let X, /z, v, p be any four screws reciprocal to the cylindroid; then the single screw y, which is reciprocal to the five screws e, X, p, v, p, must lie on the cylindroid because it is reciprocal to X, p, v, p, and therefore y is the screw required. The solution must generally be unique, for if a second screw were reciprocal to e, then the whole cylindroid would be reciprocal to e; but this is not the case unless e fulfil certain conditions (§ 22). 27. Properties of the Cylindroid*. We enunciate here a few properties of the cylindroid for which the writer is principally indebted to that accomplished geometer the late Dr Casey. The ellipse in which a tangent plane cuts the cylindroid has a circle for its projection on a plane perpendicular to the nodal line, and the radius of the circle is the minor axis of the ellipse. The difference of the squares of the axes of the ellipse is constant wherever the tangent plane be situated. The minor axes of all the ellipses lie in the same plane. The line joining the points in which the ellipse is cut by two screws of equal pitch on the cylindroid is parallel to the major axis. The line joining the points in which the ellipse is cut by two intersecting screws on the cylindroid is parallel to the minor axis For some remarkable quaternion investigations into “ the close connexion between the theory of linear vector functions and the theory of screws” see Professor C. J. Joly, Trans. Royal Irish Acad., Vol. xxx. Part xvi. (1895), and also Proc. Royal Irish Acad., Third Series, Vol. v. No. 1, p. 73 (1897).