A Treatise on the Theory of Screws

190 THE THEORY OF SCREWS. [190- shall acquire a given kinetic energy E, in consequence of the impulsive wrench. We have from § 91 the equation F 1 .2 M uj We can assign a geometrical interpretation to this equation, which will lead to some interesting results. Through the centre 0 of the pitch quadric the plane A reciprocal to y is to be drawn. A sphere (§ 181) is to be described touching the plane A at the origin 0, the diameter of the sphere being so chosen that the intercept OP made by the sphere on a radius vector parallel to any screw 0 is equal to (§ 181). The quantity ue is inversely proportional to the radius vector OQ of the ellipsoid of inertia, which is parallel to 0 (§ 186). Hence for all the screws of the screw system which acquire a given kinetic energy in consequence of a given impulse, we must have the product OP. OQ constant. From a well-known property of the sphere, it follows that all the points Q must lie upon a plane A', parallel to A. This plane cuts the ellipsoid of inertia in an ellipse, and all the screws required must be parallel to the generators of the cone of the second degree, formed by joining the points of this ellipse to the origin, 0. Since we have already shown how, when the direction of a screw belonging to a screw system of the third order is given, the actual situation of that screw is determined (§ 180), we are now enabled to ascertain all the screws 0 on which the body acted upon by a given impulse would acquire a given kinetic energy. The distance between the planes A and A' is proportional to OP. OQ, and therefore to the square root of E. Hence, when the impulse is given, the kinetic energy acquired on a screw determined by this construction is greatest when A and A' are as remote as possible. For this to happen, it is obvious that A will just touch the ellipsoid of inertia. The group of screws will, therefore, degenerate to the single screw parallel to the diameter of the ellipsoid of inertia conjugate to A. But we have seen (§ 130) that the screw so determined is the screw which the body will naturally select if permitted to make a choice from all the screws of the system of the third order. We thus see again what Euler’s theoi-em (§ 94) would have also told us, viz., that when a quiescent rigid body which has freedom of the third order is set in motion by the action of a given impulsive wrench, the kinetic energy which the body acquires is greater than it would have been had the body been restricted to any other screw of the system than that one which it naturally chooses.