A Treatise on the Theory of Screws

203] PLANE REPRESENTATION OJ? THE THIRD ORDER. 203 we shall obtain y (Pa — Z>) (j?3 — p) (p3-Pi)i(Pi-Pi)i’ , = _ (Pi-PHPi-P) J (Pi-P-^^-p^’ 2' = - (P1-P)(P2~P) {Pi-P^^-Pl)1' and as these values satisfy the equation of the quadric, the theorem has been proved. The family of quadric surfaces are therefore inscribed in a common tetra- hedron, and they have four common points, as well as four common tangent planes. For, write the two cones Æ2 + 2/2 + Ä2 = 0, p.x'2 + piy-+p-iz‘= 0. These cones have four generators in common, and the four points in which these generators cut the plane at infinity will lie on every surface of the type {pi - p) & + (P2 -p)ya+ {Pi -p)** (pi - p) (P2 -p)(ps~p}=0- We now see the distribution of the screws in the imaginary planes. In each one of these planes there are a system of parallel lines; each line of this system passes through the same point at infinity, which is, of course, one of the four points just referred to. Every line of the parallel set, when it receives appropriate pitch, belongs to the three-system. It thus appears that the ambiguity in the pitches of the screws in the planes is only apparent. The system of screw co-ordinates which usually defines a screw with absolute definiteness, loses that definiteness for the screws in these planes. Each plane contains a whole pencil of screws, radiating from a point at infinity, but the co-ordinates can only represent these screws collectively, for the three co-ordinates then represent, not a single screw, but a whole pencil of screws. As the pitches vary on every screw of the pencil, the co-ordinates can only meet this difficulty by representing the pitch as indeterminate. The proof that only a single screw of each pitch is found in the pencil is easily given. If there were two, then the same hyperboloid would have two generators in this plane of equal pitch ; but this is impossible, because, from the known properties of the three-system, only one of these generators belongs to the three-system, and the other to the reciprocal system.