A Treatise on the Theory of Screws

168 THE THEORY OF SCREWS. [167, Eliminating 0, and omitting the accents, we have, as the equation of the cubic, yx* + ma? sin 2y + l2y + 2lmx cos 2y - ml2 sin 2y = 0. The chord joining the two points 0 and 0' has, as its equation, mx cos (0 + O') [cos (2y — 0 — O') + cos (0 — 0')] + ly — ml sin (2y — 0-0') cos (0 + 0') - ml cos (0 — 0') sin (0 + O') = 0. If this be the chord joining the screws of equal pitch, then 0 + 0' = 0, and the equation reduces to mx (cos 2y + cos 20) + ly - ml sin 2y - 0. We thus see that this chord, which in the general section envelops a parabola, now passes constantly through the fixed point x — 0, y = + m sin 2y. This result could have been foreseen; for, consider that screw on the cylin- droid (and there must always be one) normal to the plane which it intersects at a point P, any ray in the plane through P is perpendicular to this screw, and, therefore, by a well-known property of the cylindroid, must intersect the curve again in two points of equal pitch. This point P is, of course, the point whose existence we have demonstrated above. 168. Relation between Two Conjugate Screws of Inertia. We have found the relation between a pair of conjugate screws of inertia so important in the dynamical part of the theory, that it is worth while to investigate the properties of the chord joining two such points in the central section. It can readily be shown that this chord must envelop a conic. This conic and the point P on the cubic through which all reciprocal chords will pass, will enable the impulsive screw, corresponding to any instantaneous screw, to be immediately determined. For, draw through any point $ that tangent to the conic which gives S as one of the two conjugate screws of inertia which must lie upon it; let S' be the other conjugate screw; then the chord PS' will cut the cubic again in the required impulsive screw. The two principal screws of inertia are found by drawing from P that tangent to the conic which has not P as one of the two conjugate screws of inertia. The two intersections of this tangent, with the cubic, are the required principal screws of inertia. We can also determine the relation between the impulsive screw and the instantaneous screw with regard to any section whatever. We have here