A Treatise on the Theory of Screws

___________ 408 _______________ _______________________ ______ __________ ________________________________ THE THEORY OF SCREWS. [368- whence we deduce the following theorem:— If the position of a mass-chain be referred to co-reciprocal screw-chains of reference, then _ 1 dT 1 dT Pi ddi pn ddf are the co-ordinates of the restraining wrench-chain which would coerce the mass-chain into continuing to twist about the same screw-chain 6*. 369. Physical meaning of the Vanishing Emanant. We may verify this theorem by the following method of viewing the subject. It must be possible to coerce the system to twist about 0 by the imposition of special constraints. The reactions of these constraints will constitute, in fact, the restraining wrench-chain. It is, however, a character- istic feature that, as the system is, ex hypothesi, still at liberty to twist about 6, the reaction of any constraints which are consistent with this freedom must lie on a screw-chain reciprocal to 0. The condition that two screw-chains, 6 and v, shall be reciprocal (§ 354) is 4" PidiPi, .,. + pn0nyn — 0, but this is clearly satisfied if for t/j, ... we substitute 1 dT _ J. dT . pn d0n Pi dOf for the equation then becomes 0dT + ø dT -n 91 dør ••• + øndøn'-°’ which, when multiplied by 0, reduces to the known identity g a dT _ . 91 dør'" + endef~Q- We thus obtain a physical meaning of this equation. It is no more than an expression of the fact that the restraining wrench-chain must be reciprocal to the instantaneous screw-chain. 370. A displacement without change of Energy. It should also be noticed that provided the twist velocities remain un- altered the kinetic energy will be unchanged by any small displacement of the mass-chain arising from a twist on any screw-chain reciprocal to the restraining screw-chain. " A particular case of this, or what is equivalent thereto, is given in Williamson and Tarleton’s Dynamics, 2nd ed., p. 432. ________