A Treatise on the Theory of Screws

160 THE THEORY OF SCREWS. [162- The line at infinity on the plane of z is of course intersected by all the real generators of the cylindroid, inasmuch as they are parallel to z. Ihe points at infinity on the planes x + iy = 0 are each the residence of an imaginary screw, also belonging to the surface. The pitches of both these screws are infinite. We may deduce the two screws of infinite pitch on the surface in another way. The equations of a screw are y — x tan 0, z = m sin 20, while the pitch is p„ + m cos 20. If tan 0 be either + i, we find both z infinite and the pitch infinite. We thus see that through the infinitely distant point I, on the nodal line of the cylindroid, two screws belonging to the surface can be drawn, just as at any finite point. The peculiarity of the two screws through I is, that their pitches are equal, i.e. both infinite, and this is not the case with any other pair of intersecting screws. It is now obvious why the envelope just considered turned out to be a parabola rather than any other conic section. Every plane section will have the line at infinity for a transversal cutting two screws of equal pitch; the envelope of such transversals must thus have the line at infinity for a tangent, i.e. must be a parabola. 163. Chord joining Two Points. If 0' and 0" be the angles by which two points on the cubic are defined, then the equation to the chord joining those points is Ax + By + <7=0; where A = 2 m cos (#' — a) cos (0" — a) cos (0' + 0"), B = h sin ß + m cos ß cos (0' + 0") sin (2a - 0' - 0") — m cos ß sin (0' + 0") cos (0' — 0"), C = — tan ß (A — m cot ß sin 2Ö') (A — m cot ß sin 20"). If in these expressions we make 0' + 0" = 0, we obtain the equation for the chord joining screws of equal pitch, as already obtained. We shall find that, in particular sections, these expressions become con- siderably simplified. Suppose, for example, that the plane of section be a tangent plane to the cylindroid. The cubic then degenerates to a straight line and a conic. The condition for this will be obvious from the equation