A Treatise on the Theory of Screws

164] THE GEOMETRY OF THE CYLINDROID. 161 of the cubic. If the coefficient of become zero, the required decomposition takes place, for y is then a factor. The necessary and sufficient condition for the plane of section being a tangent, is therefore m sin 2a + h tan ß — 0. When this is the case, the three expressions of A, B, C may be divided by a common factor, 2m cos (O' — a) cos (0" — a), and we have A = cos (0' + 0"), B = — cos ß sin (Ö' + 0"), C = — 2m cot ß sin (a + 0') sin (a + 0"). If the screws be of equal pitch, 0' + 0" = 0, the coefficient of y disappears, and we see that all the chords are merely lines parallel to the axis of y, which is parallel to one of the axes of the ellipse. The equation to the chord then becomes (cos 2a + cos 20) {x + m cot ß (cos 2a — cos 20)j = 0. For a given value of x there are two values of 0 corresponding to the two chords that can be drawn through the point. One of these chords is parallel to y, and has a 0 obtained from the equation x + m cot ß (cos 2a - cos 20) = 0. 7T The other value of 0 is „ — a, from the equation cos 2a + cos 20 = 0. This is independent of x, as might have been foreseen from the fact that the two screws of equal pitch are in this case the line in the section and the other screw of equal pitch. The latter cuts the section in a certain point, and, of course, all chords through this point meet the curve in two screws of equal pitch. 164. Reciprocal Screws. Another branch of the subject must now be considered. We shall first investigate the following general problem :— From any point, P, a series of transversals is drawn across each pair of reciprocal screws on the cylindroid. It is required to determine the cone which is the locus of these transversals. We shall show that this is a cone of the second degree. B. U