A Treatise on the Theory of Screws

293] DEVELOPMENTS OF THE DYNAMICAL THEORY. 311 disposable quantities in the choice of 77, and five more in the choice of f. We ought, therefore, to have ten disposable co-ordinates for the designing and the placing of the rigid body. But there are not so many. We have three for the co-ordinates of its centre of gravity, three for the direction of its principal axes, and three more for the radii of gyration. The other circumstances of the rigid body are of no account for our present purpose. It thus appears that if the four screws had been chosen arbitrarily we should have ten conditions to satisfy, and only nine disposable co-ordinates. It is hence plain that the four screws cannot be chosen quite arbitrarily. They must be in some way restricted. We can show as follows that these restrictions are not fewer than two. Draw a cylindroid A through a, ß, and another cylindroid P through 77^. Then an impulsive wrench about any screw <0 on P will make the body twist about some screw 0 on Æ As 10 moves over P, so will its corre- spondent 0 travel over A. It is shown in § 125 that any four screws on P will be equianharmonic with their four correspondents on A, and that consequently the two systems are homographic. In general, to establish the homography of two cylindroids, three cor- responding pairs of screws must be given; and, of course, there could be a triply infinite variety in the possible homographies. It is, however, a somewhat remarkable fact that in the particular homography with which we are concerned there is no arbitrary element. The fact that the rigid body is supposed quite free distinguishes this special case from the more general one of § 290. Given the cylindroids A and P, then, without any other considerations whatever, all the corresponding pairs are determined. This is first to be proved. If the mass be one unit, and the intensity of the impulsive wrench on a> be one unit, then the twist velocity acquired by 0 is (§ 280) cos (0m) Pe ’ where cos (0<a) denotes the cosine of the angle between the two screws 0 and a>, and where pe is the pitch of 0. If, therefore, pe be zero, then cos (0a>) must be zero. In other words, the two impulsive screws w1; on P, which correspond to the two screws of zero pitch 0lt 0,2 on A, must be at right angles to them, respectively. This will in general identify the correspondents on P to two known screws on A. We have thus ascertained two pairs of correspondents, and we can now determine a third pair. For if a>3 be a screw on P reciprocal to 02, then its correspondent 0S will be reciprocal to w2. Thus we have three pairs 02, 0.2, 03 on A, and their three correspondents &>2, co, on P. This establishes the