A Treatise on the Theory of Screws

202 THE THEORY OF SCREWS. [202, If we make a = (p2-jp3)^; ß = {ps-p-^', y = {p1-p^, the equations of the four planes are expressed in the form - ax + ßy + yz — aßy = 0, - ax + ßy + yz- aßy - 0, - ax — ßy + yz — aßy = 0, - ax + ßy - yz — aßy = 0. It is lemarkable thut the three equations of the axis for each of these screws here coalesce to a single one. The screw of indeterminate pitch is thus limited, not to a line, but to a plane. The same may be said of each of the other three screws of indeterminate pitch; they also are each limited to a plane found by giving variety of signs to the radicals in the equations just written. We have thus discovered that the complete locus of the screws of a three-system consists, not only of the family of quadrics, which contain the screws of real or imaginary, but definite pitch, but that it also contains a tetrahedron of four imaginary planes, each plane being the locus of one of the four screws of indefinite pitch. 203. Relation of the Four Planes to the Quadrics. The planes have an interesting geometrical connexion with the family of quadrics, which we shall now develop. The first theorem to be proved is, that each of the quadrics touches each of the planes. This is geometrically obvious, inasmuch as each quadric contains all the screws of the system which have a given pitch p - but each of the planes contains a system of screws of every pitch, among which there must be one of pitch p. There will thus be a ray in the plane, which is also a generator of the hyper- boloid—but this, of course, requires that the plane be a tangent to the hyperboloid. It is easy to verify this by direct calculation. Write the quadric, Ü»-p) x1 + (p2-p) f + -p) + (?1 -p) - p) = o. Ihe tangent plane to this, at the point x, y\ z, is (Pi-p)xx + (p2 — p) yy + (ps — p) zz' + (p1 — p) (p., —/>)(p3 — p) = 0. If we identify this with the equation ax + ßy + yz — aßy = 0,