A Treatise on the Theory of Screws

165] THE GEOMETRY OF THE CYLINDROID. 165 reciprocal to that through P; therefore the third intersection of S with the surface must coalesce with P, or, in other words, <S' must be a tangent to the surface at P. We have thus shown that in the case where two reciprocal chords through a point on the cubic coalesce into one, that one must be the tangent to the cubic at P. But the two reciprocal chords through a point will only coalesce when the point lies on the hyperbola, in which case the two chords unite into the tangent to the hyperbola. Consider, then, the case where the hyperbola meets the cubic at a point P, inasmuch as P lies on the hyperbola, the two chords coalesce into a tangent thereto, but because they do coalesce, this line must needs be also a tangent to the cubic; hence, whenever the hyperbola meets the cubic the two curves must have a common tangent. Altogether the curves meet in six points, which unite into three pairs, thus giving the required triple contact between the hyperbola and the cubic. If a constant h be added to all the pitches of the screws on a cylindroid, then, as is well known, the screws so altered still represent a possible cylin- droid (§ 18). The variations of h produce no alteration in the cubic section of the cylindroid; but, of course, the hyperbola just considered varies with each change of h. In every case, however, it has the triple contact, and there is also a fixed tangent which must touch every hyperbola. This is the chord joining the two principal screws on the cylindroid; for, as these are reciprocal, notwithstanding any augmentation to the pitches, their chord must always touch the hyperbola. The system of hyperbolæ, corresponding to the variations of h, is thus concisely represented; they must all touch this fixed line, and have triple contact with a fixed curve: that is, they must each fulfil four conditions, leaving one more disposable quantity for the complete definition of a conic. See Appendix, note 4. We write the tangent to the hyperbola or the reciprocal chord in the form L cos 2i/r + M sin 2^ + N = 0. If a pair of values can be found for x and y, which will simultaneously satisfy Z = 0, > = 0, 2V=0, then every chord of the type L cos 2-i/r + M sin + TV = 0 must pass through this point. The condition for this is, that the discriminant of the hyperbola is zero, and we find the discriminant to be sin ß (X2 - 1) (A tan ß + m sin 2a).