A Treatise on the Theory of Screws

gBøMSHMMMMHMHMMHMMHMHM 328 THE THEORY OF SCREWS. [306 ray in »S'. Hence, we see that the ray Q must lie in each of the planes so constructed, and is therefore determined. In fact, it is merely the transversal drawn from the centre of gravity to intersect both the screws of zero pitch on the cylindroid (y, %). We have thus proved that when two pairs of corresponding impulsive screws and instantaneous screws are given, we know the centre of the momental ellipsoid, we know the directions of three of its conjugate diameters, and we know the radii of gyration on two of those diameters. The radius of gyration on the third diameter remains arbitrary. Be that radius what it may, the rigid body will still fulfil the condition rendering a, i) and ß, g respective pairs of instantaneous screws and impulsive screws. We had from the first foreseen that the data would only provide eight coordinates, while the specification of the body required nine. We now learn the nature of the undetermined coordinate. It appears from this investigation that, if two pairs of impulsive screws and the corresponding instantaneous screws are known, but that if there be no other information, the rigid body is indeterminate. It follows that, if an impulsive screw be given, the corresponding instantaneous screw will not generally be determined. Each of the possible rigid bodies will have a different instantaneous screw, though the impulsive screw may be the same. It was, however, shown (§ 299), that all the instantaneous screws which pertain to a given impulsive screw lie on the same cylindroid. It is a cylindroid of extreme type, possessing a screw of infinite pitch, and degenerating to a plane. Even while the body is thus indeterminate, there are, nevertheless, a system of impulsive screws which have the same instantaneous screw for every rigid body which complies with the expressed conditions. Among these are, of course, the several screws on the impulsive cylindroid (y. %) which have each the same corresponding screw on the instantaneous cylin- droid (a, ß), whatever may be the body of the system to which the impulsive wiench is applied. But the pairs of screws on these two cylindroids are indeed no more than an infinitesimal part of the total number of pairs of screws that are circumstanced in this particular way. We have to show that there is a system of screws of the fifth order, such that an impulsive wrench on any one of those screws y will make any body of the system com- mence to twist about the same screw a. As already explained, the system of rigid bodies have a common centre of gravity. Any force, directed through the centre of gravity, will produce a linear velocity parallel to that force. This will, of course, apply to every body of the system. All possible forces, which could be applied to one point, form a system of the third order of a very specialized type; Each one