A Treatise on the Theory of Screws

175] FREEDOM OF THE THIRD ORDER. 175 Hence the pitches of all the real screws of the screw system $ are inter- mediate between the greatest and least of the three quantities pa,pf>, py. 175. Construction of a three-system from three given Screws. If a family of quadric surfaces have one pair of generators (which do not intersect) in common, then the centre of the surface will be limited to a certain locus. We may investigate this conveniently by generalizing the question into the search for the locus of the pole of a fixed plane with respect to the several quadrics. Let A be the given plane, I be the ray which joins the two points in which the given pair of generators intersect A, X be the plane through I and the first generator, F the plane through I and the second generator, B the plane through I which is the harmonic conjugate of A with respect to X and P. Then B is the required locus. For, draw any quadric through the two given generators, and let 0 be the pole of A with respect to that quadric. Draw a transversal through 0 cutting the plane A in the point A and the first and second generators in X1 and F, respectively. Since A4 is on the polar of 0 it follows that OX1A1Y1 is an harmonic section. But the transversal must be cut harmonically by the pencil of planes Z(BXAY) and hence 0 must lie in B, which proves the theorem. In the particular case when A is the plane at infinity, then 0 is the centre of the quadric. A plane parallel to the two generators cuts the plane at infinity in the line I, and the planes X, Y and B must also contain I. Then A, B, X, Y are parallel planes. Any transversal across X and Y is cut harmonically by B and A, and as A is at infinity, the transversal must be bisected ;it B. It thus appears that when a family of quadrics have one pair of non-intersecting generators in common, then the plane which bisects at right angles the shortest distance between these generators is the locus of the centres of the quadrics. If therefore three generators of a quadric are given, the three planes determined by each pair of the quadrics determine the centre by their intersection. The construction of the axes of the quadric may be effected geometrically in the following manner. Draw three transversals Q1; Q2, Q3 across the three given generators Rlt R2, Rs. Draw also two other trans- versals R4, R$ across Q4, Q.,, Q3. Construct the conic which passes through the five points in which Rlt R2, Rs, Rit R5 intersect the plane at infinity. Find the common conjugate triangle to this conic and to the circle which is the intersection of every sphere with the plane at infinity. Then the three