A Treatise on the Theory of Screws

179] FREEDOM OF THE THIRD ORDER. 181 system. Since P'RX is to be reciprocal to Pl\, it is essential that Rt be a right angle; similarly R., is a right angle. The reciprocal cylindroid, whose axis passes through P', will be identical with the cylindroid belonging to the system whose axis passes through P; but the two will be differently posited. If the angle at P be a right angle, the points rJ\ and T2 are at infinity; therefore, the plane touches the quadrics at infinity; it must, therefore, touch the asymptotic cone, and must, therefore, pass through the centre of the pitch quadric 0; but P is the centre of the cylindroid in this case, and, therefore, the centre of the cylindroid must lie in the plane A. The position of the centre of the cylindroid in the plane A is to be found by the following construction :—Draw through the centre 0 a diameter of the pitch quadric conjugate to the plane A. Let this line intersect the pitch quadric in the points Plt P2, and let S, S' (Fig. 38) be the feet of the perpendiculars let fall from Plt P2 upon the plane A. Draw the asymptotes OL, OM to the section of the pitch quadric, made by the plane A. Through S and S' draw lines in the plane A, ST, ST', S'T, S'T’, parallel to the asymptotes, then T' and T are the centres of the two required cylindroids which belong to the two reciprocal screw systems. Fig. 38. This construction is thus demonstrated :— The tangent planes at P2, P2 each intersect the surface in lines parallel to OL, OM. Let us call these lines P2L2, P2M2 through the point Plt and P2L2, P2M2 through the point P2. Then P^, P2M2 are screws belonging to the system, and P1M1, P2L2 are reciprocal screws. Since OL is a tangent to the pitch quadric, it must pass through the intersection of two rectilinear generators, which both lie in a plane which contains OL; but since OL touches the pitch quadric at infinity, the two generators in question must be parallel to OL, and therefore their projections on the plane of A must be S'T, ST'. Similarly for ST, S'T'; hence ST' and S'T' are the projections of two screws belonging to the system, and therefore the centre of the cylindroid is at T'. In a similar way it is proved that the centre of the reciprocal cylindroid is at T. Having thus determined the centre of the cylindroid, the remainder of the construction is easy. The pitches of two screws on the surface must be proportional to the inverse square of the parallel diameters of the section of the pitch quadric made by A. Therefore, the greatest and least pitches will be on screws parallel to the principal axes of the section. Hence, lines