A Treatise on the Theory of Screws

176 THE THEORY OF SCREWS. [175- rays from the centre of the quadric to the vertices of this triangle are the three principal axes of the quadric. We thus prove again that if a and ß be any two screws of a three-system, the centre of the pitch-quadric must lie in the principal plane of the cylindroid through a and ß. For the common perpendicular to any two screws of equal pitch on the cylindroid will be bisected by the principal plane and therefore any hyperboloid through these two screws of equal pitch must have its centre in that plane. 176. Screws through a Given Point. We shall now show that three screws belonging to S, and also three screws belonging to S', can be drawn through any point x', y', z'. Substitute x', y', z', in the equation of § 175 and we find a cubic for k. This shows that three quadrics of the system can be drawn through each point of space. The three tangent planes at the point each contain two generators, one belonging to S, and the other to S'. It may be noticed that these three tangent planes intersect in a straight line. From the form of the equation it appears that the sum of the pitches of three screws through a point is constant and equal to pa +pß +py. Two intersecting screws can only be reciprocal if they be at right angles, or if the sum of their pitches be zero. It is hence easy to see that, if a sphere be described around any point as centre, the three screws belonging to /S', which pass through the point, intersect the sphere in the vertices of a spherical triangle which is the polar of the triangle similarly formed by the lines belonging to S'. We shall now show that one screw belonging to 6' can be found parallel to any given direction. All the generators of the quadric are parallel to the cone (pa — k) x2 + (pß — k)y2 + (py - k)z* = 0, and k can be determined so that this cone shall have one generator parallel to the given direction; the quadric can then be drawn, on which two gene- rators will be found parallel to the given direction; one of these belongs to *S', while the other belongs to S'. It remains to be proved that each screw of S has a pitch which is propor- tional to the inverse square of the parallel diameter of the pitch quadric*. * This theorem is connected with the linear geometry of Pliicker, who has shown (Neue Geometrie des Raumes, p. 130) that fc1x2 + i2?/2 + fc3z2 + Z;1A:2fc3=01 is the locus of lines common to three linear complexes of the first degree. The axes of the three complexes are directed along the co-ordinate axes, and the parameters of the complexes are klt k„, k3; the same author has also proved that the parameter of any complex belonging to the “dreigliedrige Gruppe” is propor- tional to the inverse square of the parallel diameter of the hyperboloid.