A Treatise on the Theory of Screws

no THE THEORY OF SCREWS. [121- 121. Displacement of a Point. Let P be a point, and let a, ß be any two screws upon a cylindroid. If a body to which P is attached receive a small twist about a, the point P will be moved to P'. If the body receive a small twist about ß, the point P would be moved to P". Then whatever be the screw 7 on the cylindroid about which the body be twisted, the point P will still be displaced in the plane PPP". For the twist about 7 can be resolved into two twists about a and ß, and therefore every displacement of P must be capable of being resolved along PP' and PP". Thus through every point P in space a plane can be drawn to which the small movements of P, arising from twists about the screws on a given cylindroid are confined. The simplest construction for this plane is as followsDraw through the point P two planes, each containing one of the screws of zero pitch; the intersection of these planes is normal to the required plane through P. The construction just given would fail if P lay upon one of the screws of zero pitch. The movements of P must then be limited, not to a plane, but to a line. The line is found by drawing a normal to the plane passing through P, and through the other screw of zero pitch. We thus have the following curious property due to M. Mannheim*, viz., that a point in the rigid body on the line of zero pitch will commence to move in the same direction whatever be the screw on the cylindroid about which the twist is imparted. This easily appears otherwise. Appropriate twists about any two screws, a and ß, can compound into a twist about the screw of zero pitch X, but the twist about X cannot disturb a point on X. Therefore a twist about ß must be capable of moving a point originally on X back to its position before it was disturbed by ct. Therefore the twists about ß and ot must move the point in the same direction. 122. Properties of the Pitch Conic. Since the pitch of a screw on a cylindroid is proportional to the inverse square of the parallel diameter of the pitch conic (§ 18), the asymptotes must be parallel to the screws of zero pitch; also since a pair of reciprocal screws are parallel to a pair of conjugate diameters (§ 40), it follows that the two screws of zero pitch, and any pair of reciprocal screws, are parallel to the rays of an harmonic pencil. If the pitch conic be an ellipse, there * Journal de Vécole Polytechnique, T. xx. cah. 43, pp. 57—122 (1870).