A Treatise on the Theory of Screws

323] THE GEOMETRICAL THEORY. 351 of corresponding impulsive screws and instantaneous screws, all the corre- sponding pairs are determined. There is no arbitrary element in the correspondence. There is no possible rigid body which would give any different correspondence. If we are given two systems of the fourth order of corresponding impulsive screws and instantaneous screws then the essential geometrical conditions (§ 281), not here making any restriction necessary, we can select one pair of correspondents arbitrarily in the two systems, and find one rigid body to fulfil the requirements. If we are given two systems of the fifth order of corresponding impulsive screws and instantaneous screws then subject to the observance of the geo- metrical conditions we can select two pairs of correspondents arbitrarily in the two systems, and find one rigid body to fulfil the requirements. If we are given two systems of the sixth order of corresponding impul- sive screws and instantaneous screws then subject to the observance of the geometrical conditions we can select three pairs of correspondents arbitrarily in the two systems, and find one rigid body to fulfil the requirements. The last paragraph is, of course, only a different way of stating the results of § 307. 323. Two Rigid Bodies. We shall now examine the circumstances under which pairs of impulsive and instantaneous screws are common to two, or more, rigid bodies. The problem before us may, perhaps, be most clearly stated as follows Let there be two rigid bodies, M and M'. If M be struck by an impulsive wrench on a screw 0, it will commence to twist about some screw X. If M' had been struck by an impulsive wrench on the same screw 0, the body would have commenced to twist about some screw which would of course be generally different from X. If 0 be supposed to occupy different positions in space (the bodies remaining unaltered), so will X and /z move into corre- spondingly various positions. It is proposed to inquire whether, under any circumstances, 0 could be so placed that X and /x should coincide. In other words, whether both of the bodies, M and M', when struck with an impulsive wrench on 0, will respond by twisting about the same instantaneous screw. It is obvious, that there is at least one position in which 0 fulfils the required condition. Let G1 G2 be the centres of gravity of M and M'. Then a force along the ray Gx G.,, if applied either to M or to M‘, will do no more than produce a linear velocity of translation parallel thereto. Hence it