A Treatise on the Theory of Screws

162 THE THEORY OF SCREWS. [164, Let a, ß, 7 be the co-ordinates of the point. Then the plane through this point, and the generator of the cylindroid defined by the equations y = æ tan 3, z — m sin 20, 15 (y — x tan 0) (y — m sin 23) = (9 — a tan 3) (z — m sin 23); or , if we arrange in powers of tan 3, we obtain A tan3 3 + B tan2 3 + G tan 3 + D = 0, in which A = az — yx; I) = yy — ßz, B — yy — ßz + 2mx — 2ma ; G = az — yx + 2mß — 2my. If the same transversal also crosses the generator defined by 3', then, A tan3 3' + B tan2 3' + G tan 3' + D = 0. When the two screws defined by 3 and 3' are reciprocal, tan 3 tan 3' = H, when II is a constant. By eliminating and rejecting the factor tan 3 — tan 3' we obtain - AW + 71 GIB - BDH + = 0. And as this is of the second degree in x, y, z, the required theorem has been proved. All these cones must pass through the centre of the cylindroid, inasmuch as the two principal screws of the cylindroid are reciprocal. If a constant be added to the pitches of all the screws on the cylindroid, then the pairs of reciprocals alter, inasmuch as II alters. The cone changes accordingly, and thus there would be through each point a family of cones, all of which, however, agree in having, as a generator, the ray from the vertex to the centre of the cylindroid. Thus, even when the cylindroid is given, we must further have the pitch of a stated screw given before the cone becomes definite. This state of things may be contrasted with that presented by the cone of reciprocal screws which may be drawn through a point. The latter depends only upon the cylindroid itself, and is not altered if all the pitches be modified by a constant increment.