A Treatise on the Theory of Screws

362] THE THEORY OF PERMANENT SCREWS. 401 362. A Property of the Kinetic Energy of a System. It is obvious that the mere alteration of the azimuth about a fixed axis from which a rigid body is set into rotation will not affect its kinetic energy, provided the position of the axis and the angular velocity both remain un- altered. A moment’s reflection will show that this principle may be extended to any movement whatever of a rigid body. At each instant the body is twisting about some instantaneous screw a with a twist velocity «. Let the body be stopped in a position which we call A. Let it receive a dis- placement by a twist of any amplitude about a and thus be brought to a position which we call B*. Finally, let the body be started from its new position B so as to twist again about a with the original twist velocity å, then it is plain that the kinetic energy of the body just before being stopped at the position A is the same as its kinetic energy just after it is started from the position B. Enunciated in a still more general form the same principle is as follows:— Any mass-chain in movement is necessarily twisting about some screw- chain. If we arrest the movement, displace the mass-chain to an adjacent position on the same screw-chain, and then start the mass-chain to twist again on the same screw-chain, with its original twist velocity, the kinetic energy must remain the same as it was before the interruption. This principle requires that whatever be the symbols employed, the function T, which denotes the kinetic energy, must satisfy a certain identical equation. I propose to investigate this equation, and its character will perhaps be best understood by first discussing the question with co-ordinates of a perfectly general type. We shall suppose the mass-chain has n degrees of freedom. Let the co-ordinates xlt...xn represent the position of the mass-chain, and let its instantaneous motion be indicated by x^,... xn. Let 0 be the initial position of the mass-chain, then in the time 8t it has reached the position O', whereof the co-ordinates are Xi + XiSt, ... xn + xn8t. The movement from 0 to O' must, like every possible movement of a system, consist of a twist about a screw-chain. This is a kinematical fact, wholly apart from whatever particular system of co-ordinates may have * We have supposed that the pitch of this displacement is the same as the pitch of a. This restriction is only introduced here because the constraints will generally forbid the body to make any other twist about the axis of a. If the body were quite free we might discard the restriction altogether as is in fact done later on (§ 376). h. 26