A Treatise on the Theory of Screws

168] THE GEOMETRY OF THE CYLINDROID. 169 to consider two conics connected with the cubic, viz. the reciprocal conic, which is the envelope of reciprocal chords, and the inertia conic, which is the envelope of chords of conjugate screws of inertia. We must provide a means of discriminating the two tangents from a point P on the cubic to either conic; any ray, of course, cuts the cubic in three points, of which two possess the characteristic relation. If P be one of these two, we may call this tangent the ‘ odd tangent.’ The other tangent will have, as its significant points, the two remaining intersections; leaving out P, we can then proceed, as follows, to determine the impulsive screw corresponding to P as the instantaneous screw:— Draw the odd tangent from P to the inertia conic, and from the con- jugate point thus found draw the odd tangent to the reciprocal conic. The reciprocal point Q thus found is the impulsive screw corresponding to P as the instantaneous screw. In general there are four common tangents to the two conics. Of these tangents there is only one possessing the property, that the same two of its three intersections with the cubic are the correlative points with respect to each of the conics. These two intersections are the principal screws of inertia. To determinp. the small oscillations we find the potential conic, the tangents to which are chords joining two conjugate screws of the potential (§ 100). The two harmonic screws are then to be found on one of the two common tangsnts to the two conics. It can be shown that both the inertia conic and the potential conic will, like the reciprocal conic, have triple contact with the cubic.