A Treatise on the Theory of Screws

494 THE THEORY OF SCREWS. can be drawn through it. This is borne out in the case of the two cylindroids. The two conics have but one point common, and the only quadric through both consists of their planes. The line of intersection of the planes intersects the conics in three distinct points, and hence another quadric cannot be drawn through the conics. As regards the ruled surfaces generated by the axes of a three-system which are parallel to the edges of a cone of degree m, the degree of the sui’face is evidently 3m. For the axes of the enclosing system -which meet any assumed line are parallel to the edges of a cubic cone, and there are 3m directions common to this cone and the director cone. Again the locus of the feet of perpendiculars on the generators from any point is a curve of degree 2»i which viewed from the point appears to have three multiple points of order m situated on the axes of the reciprocal three-system passing through the point. For if we take any plane and consider its intersections with the curve, we find easily that the axes of the enclosing system whose feet of perpendiculars from the point lie in the plane are parallel to the edges of a quadric cone. The theorem about the apparent multiple points follows from consideration of the cylindroids of the enclosing system whose double lines pass through the assumed point. We also note this construction for Art. 180. Assume a radius of the pitch quadric, draw the tangent plane and let fall the central perpendicular on the plane. Measure off on the radius and on the perpendicular the reciprocals of their lengths, thus determining a triangle. Through the centre draw a normal to the plane of the triangle equal in length to double the area of the triangle multiplied by the product of the three principal pitches. This is the perpendicular to the required axis if we consider rotation round the line from the perpendicular to the radius as positive. One more point may be mentioned. If we take the cone reciprocal to the director cone, that is the cone whose edges are perpendicular to the tangent planes, and if we use this new cone for selecting the generators of a ruled surface from the reciprocal three-system, the two ruled surfaces have a common line of striction and they touch one another along this line. This is the extension of the theorem that the reciprocal screw at right angles to the generators of a cylindroid coincides with the axis. NOTE VII. Note on homographic transformation, § 246. That there must be in general linear relations between the co-ordinates of the screws of an instantaneous system, and the co-ordinates of the corresponding impulsive screws is proved as follows. Let a be an impulsive screw and let ß be the corresponding instantaneous screw with respect to a free body. Let A, B, C, D, E, F be six independent screws which we shall take as screws