A Treatise on the Theory of Screws

24] RECIPROCAL SCREWS. 29 generator through V, and that, therefore, the cone drawn from 0 to the ellipse TLVM is the cone required. We hence deduce the following construction for the cone of reciprocal screws which can be drawn to a cylindroid from any point 0. Draw through 0 a line parallel to the nodal line of the cylindroid, and let T be the one real point in which this line cuts the surface. Find the second screw LM on the cylindroid which has a pitch equal to the pitch of the screw which passes through T. A plane drawn through the point / and the straight line LM will cut the cylindroid in an ellipse, the various points of which joined to 0 give the cone required*. We may further remark that as the plane TLM passes through a gene- rator it must be a tangent plane to the cylindroid at one of the intersections, suppose L, while at the point M the line LM must intersect another generator. It follows (22) that L must be the foot of the perpendicular from T upon LM, and that M must be a point upon the nodal line. 24. Locus of a Screw Reciprocal to Four Screws. Since a screw is determined by five quantities, it is clear that when the four conditions of reciprocity are fulfilled the screw must generally be confined to one ruled surface. But this surface can be no other than a cylindroid. For, suppose three screws X, /a, v, which were reciprocal to the four given screws did not lie on the same cylindroid, then any screw </> on the cylindroid (X, /x), and any screw ifr on the cylindroid (X, v) must also fulfil the conditions, and so must also every screw on the cylindroid (</>, yfr) (22). We should thus have the screws reciprocal to four given screws, limited not to one surface, as above shown, but to any member of a family of surfaces. The construction of the cylindroid which is the locus of all the screws reciprocal to four given screws, may be effected in the following manner: Let a, ß, 7, 8 be the four screws, of which the pitches are in descending order of magnitude. Draw the cylindroids (ot, y) and (/3, 8). It <r be a lineal magnitude intermediate between pß and py, it will be possible to choose two screws of pitch cr on (ot, y), and also two screws of pitch a on (/?, 3). Diaw the two transversals which intersect the four screws thus selected; attribute to each of these transversals the pitch - a, and denote the screws thus pro- duced by 3, <}>■ Since intersecting screws are reciprocal when the sum of then- pitches is zero, it follows that 3 and </> must be reciprocal to the cylindroids (a, 7) and (#, 3). Hence all the screws on the cylindroid (3, </>) must be re- ciprocal to a, ß, y, 8, and thus the problem has been solved. * M. Appell has proved conversely that the cylindroid is the only conoidal surface for which the feet of the perpendiculars from any point on the generators form a plane curve. Revue de Mathématiques Spéciales, V. 129-30 (1895). More generally we can prove that this property cannot belong to any ruled surface whatever except a cylindroid and of course a cylinder.