A Treatise on the Theory of Screws

172 THE THEORY OF SCREWS. [172- competent to execute, it will be necessary to examine the mutual connexion of the doubly infinite number of screws which form the screw system. It will be most convenient in the first place to classify the screws in the system according to their pitches; the first theorem to be proved is as follows:— A ll the screws of given pitch + k in a three-system lie upon a hyperboloid of which they form one system of generators, while the other system of gene- rators with the pitch — k belong to the reciprocal screw system. This is proved as follows:—Draw three screws, p, q, r, of pitch + k belonging to Draw three screws, I, m, n, each of which intersects the three screws p, q, r, and attribute to each of I, m, n, a pitch — k. Since two intersecting screws of equal and opposite pitches are reciprocal, it follows that p, q, r, must all be reciprocal to I, m, n. Hence, since the former belong to S, the latter must belong to S'. Every other screw of pitch + k intersecting I, m, n, must be reciprocal to S', and must therefore belong to & But the locus of a straight line which intersects three given straight lines is a hyperboloid of one sheet, and hence the required theorem has been proved. 173. The Pitch Quadric. One member of the family of hyperboloids obtained by varying k presents exceptional interest. It is the locus of the screws of zero pitch belonging to the screw complex. As this quadric has an important property (§176) besides that of being the locus of the screws of zero pitch, it is desirable to denote it by the special phrase pitch quadric. We shall now determine the equation of the pitch quadric. Let one of the principal axes of the pitch quadric be denoted by x, this will intersect the surface in two points through each of which a pair of generators can be drawn. One generator of each pair will belong to S, and the other to S'. Each pair of generators will be parallel to the asymptotes of the section of the pitch quadric by the plane containing the remaining principal axes y and z. Let p, v be the two generators belonging to S, then lines bisecting internally and externally the angle between two lines in the plane of y and z, parallel to p, v will be two of the principal axes of the pitch quadric. Draw the cylindroid (pv). The two screws of zero pitch on the cylindroid are equidistant from the centre of the cylindroid, and the two rectangular screws of the cylindroid bisect internally and externally the angle between the lines parallel to the screws of zero pitch. Hence it follows that the two rectangular screws of the cylindroid (pv) must be on the axes of y and z of the pitch quadric. We shall denote these screws by ß and y, and their