A Treatise on the Theory of Screws

190] FREEDOM OF THE THIRD ORDER. 189 188. Lemma. If from a screw system of the ?ith order we select n screws Alt , A„, which are conjugate screws of inertia (§ 87), and if be any screw which is reciprocal to A2,...,An, then an impulsive wrench on will cause the body, when only free to twist about the screws of the system, to commence to twist about A2. Let B, be the screw which, if the body were perfectly free, would be the impulsive screw corresponding to as the instantaneous screw. must be reciprocal to A2,...,An (§ 81). Take also 6 — n screws of the reciprocal system Br,...,B6_n. Then the 8 — n screws Rlt Slt B^,..., B6_n must be reciprocal to the n— 1 screws A2, ...,An, and therefore the 8 — n screws must belong to a screw system of the (7 — n)th order. Hence an impulsive wrench upon the screw can be resolved into components on Rlt Bt,... ,A-w Of these all but the first are neutralised by the reactions of the constraints, and by hypothesis the effect of an impulsive wrench upon jRj is to make the body commence to twist about and therefore an impulsive wrench on would make the body twist about A2. 189. Relation between the Impulsive Screw and the Instan- taneous Screw. A quiescent rigid body which possesses freedom of the third order is acted upon by an impulsive wrench about a given screw It is required to determine the instantaneous screw 0, about which the body will commence to twist. The screws which belong to the system, and are at the same time reci- procal to y, must all lie upon a cylindroid, as they each fulfil the condition of being reciprocal to four screws. All the screws on the cylindroid are parallel to a certain plane drawn through the centre of the pitch quadric, which may be termed the reciprocal plane with respect to the screw y. The reciprocal plane having been found, the diameter conjugate to this plane in the ellipsoid of inertia is parallel to the required screw 0. For let p and v denote two screws of the system parallel to a pair of conjugate diameters of the ellipsoid of inertia in the reciprocal plane. Then 0, p,, v are a triad of conjugate screws of inertia; but y is reciprocal to y and v, and, therefore, by the lemma of the last article, an impulsive wrench upon y will make the body commence to twist about 6. 190. Kinetic Energy acquired by an Impulse. We shall now consider the following problem:—A quiescent rigid body of mass M receives an impulsive wrench of intensity y" on a screw y. We have now to determine the locus of a screw 0 belonging to a screw system of the third order, such that, if the body be constrained to twist about 0, it