A Treatise on the Theory of Screws

 400 THE THEORY OF SCREWS. [360- which expresses the freedom of the body (§ 96). It may sometimes appear that the intensity of the necessary wrench on p vanishes. The body in such a case requires no coercion beyond that of the original constraints to preserve 6 as the screw about which it twists, and when this is the case we shall describe 6 as a permanent screw. This use of the word permanent does not imply that the body could remain for ever twisting about this screw, for the movement of the body to an appreciable distance will in general entail some change in its relation to the constraints. The character- istic of the permanent screw is the absence of any acceleration in the body twisting about it, using the word acceleration in its widest sense. In the earlier parts of the chapter we shall discard the restrictions involved in the assumption that the material arrangement is only a single rigid body. The doctrine of screw-chains (Chap. XXIV.) enables us to extend a considerable portion of the present theory to any mass-chain whatever. Any number of material parts connected in any manner must still conform to the general law, that the instantaneous movements can always be repre- sented by a twist about a certain screw-chain. In general the mass-chain will have a tendency to wander from twisting about the original screw-chain. In such cases the position of the instantaneous screw-chain cannot be maintained without the imposition of further coercion than that which the constraints supply. This additional set of forces may be applied by a restraining wrench-chain, the relation of which to the instantaneous screw- chain we shall have to consider. Sometimes it may appear that no restraining wrench-chain is necessary beyond one of those provided by the reaction of the constraints. The instantaneous screw-chain is then to be described as permanent. 361. Different properties of a Principal Axis. Another preliminary matter should be also noticed, because it exhibits the relation of the subject discussed in this chapter to some other parts of the Theory of Screws. In the ordinary theory of the rigid body there are, as is well known, two distinct properties of a principal axis which possess dynamical significance. We may think of a principal axis as the axis of a couple which, when applied impulsively to the body, will set it rotating about this axis. We may also think of the principal axis as a direction about which, if a body be once set in rotation, it will continue to rotate. The first of these properties by suitable generalization opens up the theory of principal screw-chains of inertia, which we have already explained in previous chapters. It is from the other property of the principal axis that the present investigation takes its rise. It is important to note that two quite different departments in the Theory of Screws happen to coalesce in the very special case of a rigid body rotating around a point.