A Treatise on the Theory of Screws

12 THE THEORY OF SCREWS. [8 But the condition that the forces shall be defined, when the position is given, is still not sufficiently precise. We might include, in this restricted group, forces which could have no existence in nature. We shall, therefore, add the condition that the system is to be one in which the continual creation of energy is impossible. An important consequence of this restriction is stated as follows:—The quantity of energy necessary to compel the body M to move from the position J. to the position B, is independent of the route by which the change has been effected. Let L and M be two such routes, and suppose that less energy was required to make the change from A to B via L than via M. Make the change viå L, with the expenditure of a certain quantity of energy, and then allow the body to return via M. Now, since at every stage of the route M the forces acting on the body are the same whichever way the body be moving, it follows, that in returning from B to A. via M, the forces will give out exactly as much energy as would have been required to compel the body to move from A to B via. M; but by hypothesis this exceeds the energy necessary to make the change via L, and hence, on the return of the body to A, there is a clear gain of a quantity of energy, while the position of the body and the forces are the same as at first. By successive repetitions of the process an indefinite quantity of energy could be created from nothing. This being contrary to experience, compels us to admit that the quantity of energy necessary to force the body from A to B is independent of the route followed. It follows that the amount of work done in a number of twists against a wrench is equal to the work that would be done in the resultant twist. For, the work done in producing a given change of position is independent of the route. We may calculate the work done in a twist against a wrench by deter- mining the amount of work done against three forces which are equivalent to the wrench, in consequence of the movements of their points of application which are caused by the twist. We shall assume the two lemmas—1st. The work done in the displace- ment of a rigid body against a force is the same at whatever point in its line of application the force acts. 2nd. The work done in the displacement of a point against a number of forces acting at that point, is equal to the work done in the same displacement against the resultant force. The theorem to be proved is as followsThe amount of work done in a given twist against a number of wrenches, is equal to the work done in the same twist against the resultant wrench.