A Treatise on the Theory of Screws

CHAPTER XVI. FREEDOM OF THE FOURTH ORDER. 212. Screw System of the Fourth Order. The most general type of a screw system of the fourth order is exhibited by the set of screws which are reciprocal to an arbitrary cylindroid (§ 75). To obtain certain properties of this screw system it is, therefore, only necessary to re-state a few results already obtained. All the screws which belong to a screw system of the fourth order and which can be drawn through a given point are generators of a certain cone of the second degree (§ 23). All the screws of the same pitch which belong to a screw system of the fourth order must intersect two fixed lines, viz. those two screws which, lying on the reciprocal cylindroid, have pitches equal in magnitude but opposite in sign to the given pitch (§ 22). One screw of given pitch and belonging to a given screw system of the fourth order can be drawn through each point in space (§ 123). As we have already seen that two screws belonging to a screw system of the third order can be found in any plane (§178), so we might expect to find that a singly infinite number of screws belonging to a screw system of the fourth order can be found in any plane. We shall now prove that all these screws envelope a parabola. A theorem equivalent to this has been already proved in a different manner in § 162. Take any point P in the plane, then the screws through P reciprocal to the cylindroid form a cone of the second order, which is cut by the plane in two lines. Thus two screws belonging to a given screw system of the fourth order can be drawn in a given plane through a given point. But it can be easily shown that only one screw of the system parallel to a given line can be found in the plane. Therefore from the point at infinity only a single finite tangent to the curve can be drawn. Therefore the other