A Treatise on the Theory of Screws

306] THE GEOMETRICAL THEORY. 327 cylindroid. For any impulsive wrench on (77, f) can be decomposed into impulsive wrenches on rj and The first of these will generate a twist velocity about a. The second will generate a twist velocity about ß. Ihese two can only compound into a twist velocity about some other screw on the cylindroid (a, /3). This must, therefore, be the instantaneous screw corre- sponding to the original impulsive wrench on (»;, £)• It is a remarkable point about this part of our subject that, as proved in § 293, we can now, without any further attention to the rigid body, corre- late definitely each of the screws on the instantaneous cylindroid with its correspondent on the impulsive cylindroid. We thus see how, from our knowledge of two pairs of correspondents, we can construct the impulsive screw on the cylindroid (77, £) corresponding to every screw on the cylindroid (st, /3). It has been already explained in the last article how a single known pair of corresponding impulsive and instantaneous screws suffice to point out a diameter of the momental ellipsoid, and also give its radius of gyration. A second pair of screws will give another diameter of the momental ellipsoid, and these two diameters give, by their intersection, the centre of gravity. As we have an infinite number of corresponding pairs, we thus get an infinite number of diameters, all, however, being paiallel to the principal plane of the instantaneous cylindroid. The radius of gyration on each of these diameters is known. Thus we get a section S of the momental ellipsoid, and we draw any pair of conjugate diameters in that section. These diameters, as well as the radius of gyration on each of them, are thus definitely fixed. When we had only a single pair of corresponding impulsive and instan- taneous screws, we could still determine one ray in the conjugate plane to the diameter parallel to the instantaneous screw. Now that we have further ascertained the centre of gravity, the conjugate plane to the diameter, parallel to the instantaneous axis, is completely determined. Every pair of corresponding impulsive and instantaneous screws will give a conjugate plane to the diameter parallel to the instantaneous screw. Ihus we know the conjugate planes to all the diameters in the plane Ä All these planes must intersect, in a common ray Q, which is, of course, the conjugate direction to the plane S. This ray Q might have been otherwise determined. Take one of the two screws, of zero pitch, in the impulsive cylindroid (y, £). Then the plane, through this screw and the centre of gravity, must, by Poinsot’s theorem already referred to, be the conjugate plane to some straight line in 8. Similarly, the plane through the centre of gravity and the other screw of zero pitch, on the cylindroid (V, £), will also be the conjugate plane to some