A Treatise on the Theory of Screws

 334 THE THEORY OF SCREWS. [309, It is shown (§ 137) that the points which represent the instantaneous screws, and the points which represent the corresponding impulsive screws, form two homographic systems. A well-known geometrical principle asserts (§ 146), that if each point on a circle be joined to its homographic corre- spondent, the chord will envelop a conic which has double contact with the circle. It is easily seen that, in the present case, the conic must be the hyperbola which touches the circle at the ends of the diameter XY, and has the rays AP and BQ for its asymptotes. The hyperbola is completely defined by these conditions, so that the pairs of correspondents are uniquely determined. Every tangent, 1ST, to this hyperbola will cut the circle in two points. I and S, such that $ is the point corresponding to the impulsive screw, and I is the point which marks out the instantaneous screw. We thus obtain a concise geometrical theory of the connexion between the pairs of cor- responding impulsive screws and instantaneous screws on a cylindroid which contains two of the principal screws of inertia of a free rigid body. For completeness, it may be necessary to solve the same problem when the cylindroid is defined by two principal screws of inertia lying along the same principal axis of the rigid body. It is easy to see that if, on the principal axis, whose radius of gyration was a, there lay any instantaneous screw whose pitch was pa, then the corresponding impulsive screw would be also on the same axis, and its pitch would be p, where x pa = a2. 310. Analogous Problem in a Three-system. Let us now take the case of the system of screws of the third order, which contains three of the six principal screws of inertia of a free rigid body. Any impulsive wrench, which acts on a screw of a system of the third order, can be decomposed into wrenches on any three screws of that system, and consequently, on the three principal screws of inertia, which in the present case the three-system has been made to contain. Each of these component wrenches will, from the property of a principal screw of inertia, generate an initial twist velocity of motion around the same screw. The three twist velocities, thence arising, can be compounded into a single twist velocity about some other screw of the system. We desire to obtain the geometrical relation between each such resulting instantaneous screw and the corresponding impulsive screw. It has been explained in Chap. XV. how the several points in a plane are in correspondence with the several screws which constitute a system of the third order. It was further shown, that if by the imposition of geometrical constraints, the freedom of a rigid body was limited to twisting