A Treatise on the Theory of Screws

8] TWISTS AND WRENCHES. 13 Let n wrenches, which consist of 3n forces acting at Ait...A3n, compound into one wrench, of which the three forces act at P, Q, R. The force at J.* may generally be decomposed into three forces along PAk, QAk, RAk. By the 2nd lemma the amount of work (JF) done against the 3n original forces, equals the amount of work done against the 9n components. It, therefore, appears from the 1st lemma, that W will still be the amount of work done against the 9ra components, of which 3n act at P, 3n at Q, 3n at R. Finally, by the 2nd lemma, W will also be the amount of work done by the original twist against the three resultants formed by compounding each group at P> Q, R. But these resultants constitute the resultant wrench, whence the theorem has been proved. We thus obtain the following theorem, which we shall find of great service throughout this book. If a series of twists A1,...Am, would compound into one twist A, and a series of wrenches Blt...Bn, would compound into one wrench B, then the energy that would be expended or gained when the rigid body per- forms the twist A, under the influence of the wrench B, is equal to the algebraic sum of the mn quantities of energy that would be expended or gained when the body performs severally each twist A1,...Am under the influence of each wrench B1,...Bn. We have now explained the conceptions, and the language in which the solution of any problem in the Dynamics of a rigid body may be pre- sented. A complete solution of such a problem must provide us, at each epoch, with a screw, by a twist about which of an amplitude also to be specified, the body can be brought from a standard position to the position occupied at the epoch in question. It will also be of much interest to know e instantaneous screw about which the body is twisting at each epoch, as e as its twist velocity. Nor can we regard the solution as quite complete, unless we also have a clear conception of the screw on which all the forces acting on the body constitute a wrench of which we should also know the intensity. There is one special feature which characterises that portion of Dynamics which is discussed in the present treatise. We shall impose no restrictions on the form of the rigid body, and but little on the character of the con- straints by which its movements are limited, or on the forces to which the rigid body is submitted. The restriction which we do make is that the body, while the object of examination, remains in, or indefinitely adjacent to, its original position. As a consequence of this restriction, we here make the remark that the amplitude of a twist is henceforth to be regarded as a small quantity.