A Treatise on the Theory of Screws

Ln MRNVHMMIi * 315 DEVELOPMENTS OF THE DYNAMICAL THEORY. 296] of the homographic equation is based on the fact that if X, X' be two instantaneous screws and p, p' the corresponding impulsive screws, then the formula —cos (X'p) H--------------s cos (Xp') = 2-ctaa/ cos (Xp) v cos (X p) must be satisfied. And generally it is satisfied. In the case of two cylindroids with normal planes it is however easy to show that there are certain pairs of screws for which this formula cannot obtain. For in such a case there is one screw X on A which is perpendicular to every screw on P, so that whatever be the p corresponding to X, cos (Xp) = 0. Since no other screw X' can be perpendicular to any screw on P we cannot have either & cos (X'p), or cos (X'p'), zero. Hence this equation cannot be satisfied and the argument that the homo- graphic equation defines corresponding pairs is in this case invalid. We might have explained the matter in the following manner. When the principal planes of A and P are normal there is one screw X on A which is perpendicular to all the screws on P. If therefore the two cylindroids were to be impulsive and instantaneous, there must be a screw 3 on P which corresponds to X. It can be shown in general (§ 301) that dK = Pk tan (X0) when d\ is the perpendicular from the centre of gravity on X; it follows that when (Å.Ø) = 90° we must have either pK zero or dK infinite. But of course it will not generally be the case that X happens to be one of the screws of zero pitch on A. Hence we are reduced to the other alternative d>. = infinity. This means that the centre of gravity is to be at infinity. But when the centre of gravity of the body is at infinity a remarkable consequence follows. All the instantaneous screws must be parallel. For if 6 be the impulsive wrench corresponding to X as the instantaneous screw, then we know that dx=p>. tan(X0), and that the centre of gravity lies in a right line parallel to X and distant