A Treatise on the Theory of Screws

181] FREEDOM OF THE THIRD ORDER. 183 lines. Oue of these lines is the required residence of the screw 6, while the other line, with a pitch equal in magnitude to that of d, but opposite in sign, belonging, as it does, to one of the other system of generators, is a screw reciprocal to the system. The family of quadric surfaces of constant pitch have the same planes of circular section, and therefore every plane through the centre cuts the quadrics in a system of conics having the same directions of axes. The cylindroid which contains all the screws of the screw system parallel to one of the planes of circular section must be composed of screws of equal pitch. A cylindroid in this case reduces to a plane pencil of rays passing through a point. We thus have two points situated upon a principal axis of the pitch quadric, through each of which a plane pencil of screws can be drawn, which belong to the screw system. All the screws passing through either of these points have equal pitch. The pitches of the two pencils are equal in magnitude, but opposite in sign. The magnitude is that of the pitch of the screw situated on the principal axis of the pitch quadric*. 181. Virtual Coefficients. Let p be a screw of the screw system which makes angles whose cosines are f, g, h, with the three screws of reference a, ß, y upon the axes of the pitch quadric. Then, reference being made to any six co-reciprocals, we have for the co-ordinates of p, Pi =/«i + ffßi + %> &c., &c., pe =fa6 + gßs + hye. Let g be any given screw. The virtual coefficient of p and g is + g^ßi + Draw from the centre of the pitch quadric a radius vector r parallel to p, and equal to the virtual coefficient just written; then the locus of the extremity of r is the sphere + y"“ + & = The tangent plane to the sphere obtained by equating the right-hand side of this equation to zero is the principal piano of that cylindroid which contains all the screws of the screw system which are reciprocal to g. * If a, b, c be the three semiaxes of the pitch quadric, and + d the distances from the centre, on a, of the two points in question, it appears from § 179 that d2d‘i=(a2 - b2) (a2 - c2), which shows that d is the fourth proportional to the primary semiaxis of the surface, and to those of its focal ellipse and hyperbola.