A Treatise on the Theory of Screws

214 THE THEORY OF SCREWS. [210, of course passes through the two points which he at the intersections of '■^JIll — LS = 0 and Q = 0. Hence we deduce the following result. The cylindroid through the two screws of zero pitch which intersect a corresponds to the ray whose equation is +^äaA + m^)-(A«i2 + p&? +p&*) (arf, +aA+aA)=0, a will of course intersect a third screw on this cylindroid and be perpen- dicular thereto. The cubic passes through the two points defined by o = aA + a.,e2 + as03, o = ^ + 0/ + ^, but these points are the points of contact of the pair of tangents drawn from the point a to the conic of infinite pitch. These two points in addition to the four points on the same conic defined by its intersection with the conic of zero pitch, make six known points on the cubic. If however we are given six points on a cubic and also a double point on the curve, then it is determinate. Thus we obtain the following geometrical construction. In the plane representation of a three-system we draw the conic of zero pitch C and the conic of infinite pitch U. To find the locus of the screws which intersect a given screw a, it is necessary to draw the two tangents from a to U and through the two points of contact, and also the four points common to C and U then draw that single cubic which has a double point at a. See Appendix, note 5. 211. Application to Dynamics. By the aid of the plane representation we are enabled to solve certain problems in the dynamics of a rigid body which has freedom of the third order. Let an impulsive wrench act upon a quiescent rigid body ; it is required to determine the instantaneous screw about which the body will commence to twist. It has been shown (§ 96) that the impulsive wrench, wherever situated, can generally be adequately represented by the reduced impulsive wrench on a screw of the three-system. The problem is therefore reduced to the determination of the point corresponding to the instantaneous screw, when that corresponding to the impulsive screw is known.