A Treatise on the Theory of Screws

237-239] FREEDOM OF THE SIXTH ORDER. 259 A2,...A6, by the sexiant of Ra,...J?6. With the exception of a common factor, the specific parameter of every pair of screws is therefore known, when seven corresponding screws are known. It will be shown in Chap. XXI. that three corresponding pairs are really sufficient. When seven instantaneous screws are known, and the corresponding seven impulsive screws, we are therefore enabled by geometrical construction alone to deduce the instantaneous screw corresponding to any eighth impulsive screw and vice versa. A precisely similar method of proof will give us the following theorem:— If a rigid body be in position of stable equilibrium under the influence of a system of forces which have a potential, and if the twists about seven given screws evoke wrenches about seven other given screws, then, without further information about the forces, we shall be able to determine the screw on which a wrench is evoked by a twist about any eighth screw. We may present the results of the present section in another form. We must conceive two corresponding systems of screws, of which the correspond- ence is completely established, when, to any seven screws regarded as belonging to one system, the seven corresponding screws in the other system are known. To every screw in space viewed as belonging to one system will correspond another screw viewed as belonging to the other system. Six screws can be found, each of which coincides with its correspondent. To a screw system of the nth order and with degree in one system will correspond a screw system of the nth order and with degree in the other system. We add here a few examples to illustrate the use which may be made of screw co-ordinates. 239. Theorem. When an impulsive force acts upon a free quiescent rigid body, the directions of the force and of the instantaneous screw are parallel to a pair of conjugate diameters in the momental ellipsoid. Let be the co-ordinates of the force referred to the absolute principal screws of inertia, then (§ 35) (Vi + %)2 + (Vs + Vtf + (Vs + %)2 = 1, and from (§ 41) it follows that the direction cosines of r/ with respect to the principal axes through the centre of inertia are (Vi + Vz), (v-i + Vt), (Vs + v<>)- 17—2