A Treatise on the Theory of Screws

134 THE THEORY OF SCREWS. [145, for the pairs of related screws becomes still more simplified by the theorem, that— When the system is one of dynamical involution, the chord joining an impulsive screw with its instantaneous screw passes through the pole of the homographic axis. We may take the opportunity of remarking, that dynamical involution is not confined to the system of the second order. It may be extended to a rigid body with any number of degrees of freedom, or even to any system of rigid bodies. Whenever it happens that the relation of impulsive screw and instantaneous screw is interchangeable in one case, it is interchangeable in every case. For, let ... 0n be the co-ordinates of an instantaneous screw, then ('§ 97) the corresponding impulsive screw has for co-ordinates, and if this latter were regarded as an instantaneous screw, then its impulsive screw would be but as this is to be only we must have w? _ uf _ uf P1~ P-i Pn2’ which shows that if the theorem be true for one pair, it is true for all. The conditions, of course, are, that any one of the following systems of equations be satisfied:— + uf _ + uf = + uf - Pl ~ Pi ~ Pn' 146. Another Construction for the Twist Velocity acquired by an Impulse. Reverting to the general case, we find that the chord A A' (Fig. 27) is cut by the homographic axis at T, so that the square of the acquired twist velocity is proportional to the ratio of TA to TA'. For, with the construction in § 142, draw HQ parallel to AT \ then, HQ : A'T :: H£l : A'fl, AT = Hfl 1 Ä'T A'tl x AW’