A Treatise on the Theory of Screws

428 THE THEORY OF SCREWS. [390, We also see that <7 must lie on the polar of the point a01; ß02, y03 with regard to the same conic. We thus obtain a geometrical construction by which we discover the restraining screw when the instantaneous screw is given. Two homographic systems are first to be conceived. A point of the first system, of which the co-ordinates are 0lt 02, 03, has as its correspondent a point in the second system, with co-ordinates a0lt ß0.2, y03. The three double points of the homography correspond, of course, to the permanent screws. To find the restraining screw y corresponding to a given instantaneous screw 0, we join 0 to its homographic correspondent, and the pole of this ray, with respect to the pitch conic, is the position of y. The pole of the same ray, with regard to the conic of inertia (§ 211), is the accelerator. It seems hardly possible to have a more complete geo- metrical picture of the relation between 77 and 0 than that which these theorems afford. 391. Calculation of Permanent Screws in a Three-system. When a three-system is given which expresses the freedom of a body we have seen how in the plane representation the knowledge of a conic (the conic of inertia) will give the instantaneous screw corresponding to any given impulsive screw. A conic is however specified completely by five data. The rigid body has nine co-ordinates. It therefore follows that there is a quadruply infinite system of rigid bodies which with respect to a given three-system will have the same conic of inertia. If in that three-system a be the instantaneous screw corresponding to 77 as the impulsive screw for any one body of the quadruply infinite system, then will y and a stand in the same relation to each other for every body of the system. The point in question may be illustrated by taking the case of a four- system. The screws of such a system are represented by the points in space, and the equation obtained by equating the kinetic energy to zero indicates a quadric. For the specification of the quadric nine data are necessary. This is just the number of co-ordinates required for the specification of a rigid body. If therefore the inertia quadric in the space representation be assumed arbitrarily, then every instantaneous screw corresponding to a given impulsive screw will be determined; in this case there is only a finite number of rigid bodies and not an infinite system for which the correspond- ence subsists. We thus note that there is a special character about the freedom of the fourth order which we may state more generally as follows. To establish a chiastic homography (§ 292) in an «-system requires (n — !)(» + 2)/2 data.