A Treatise on the Theory of Screws

165J THE GEOMETRY OF THE CYLINDROID. 163 The discriminant of the cubic - All1 + ACIB- BDH + D* = 0 is AW (AW + BSD + AC3 — 5’T BW -1ABCD). Omitting the factor A~D~, we have, for the envelope of the system of cones, the cone of the fourth order, found by equating the expression in the biacket to zei’o. It may be noted that the same cone is the envelope of the planes AH3 + BIB + CII + I) = 0. 165. Application to the Plane Section. We next study the chord joining a pair of reciprocal points on the cubic of § 160. Take any point in the plane of the section; then, as we have just seen, a cone of screws can be drawn through this point, each ray of which crosses two reciprocal screws. This cone is cut by the plane of section in two lines, and, accordingly, we see that through any point in the plane of section two chords can be drawn through a pair of reciprocal points. The actual situation of these chords is found by drawing a pair of tangents to a certain hyperbola. This will now be proved. The values of 0 and 0', which correspond to a pair of reciprocal points, fulfil the condition tan 0 tan 0’ = H; whence, cos (0 — 0') = X cos (0 + 0'); where, for brevity, we write X. instead of 1+tf 1 — H' If, further, we make 0 + 0' = yfr, we shall find, for the equation of the chord, Pa> + Qy+Rz = o; in which, ^22, 772/ 'TfL P = -~ (X + cos 2a) -I- sin 2a sin (X + cos 2a) cos 2i/r, Aa Ai Aa T i ■ n . m • n rø COS ß „ Q = h sm ß + cos ß sin 2a - (X + cos 2a) sm 2-A Ai Ai ‘ QTL + cos ß sin 2a cos 2i/r, '772'^ 779^ R = - h° tan ß - — (V - 1) cot ß 4- Xhm sin 2i/r - — (X2 -1) cot ß cos 2->/r. Aa 11-2