A Treatise on the Theory of Screws

359] THE THEORY OF SCREW-CHAINS. 397 is reciprocal to <f>, ß will be reciprocal to 6. This, it will be observed, is a generalization of a property of which much use has been previously made (§ 81). The proof is as follows. The co-ordinates of the instantaneous chains are Otj, ... Ä72,, ßi,...ßn- The co-ordinates of the corresponding impulsive chains are Wj2ai un2an pl Pn and Ui2ßi ™nßn Pi ’ ’" Pn If the chain a be reciprocal to the impulsive chain which produces ß, then we have UiO-ißi + ... + un2anßn = 0 ; but this being symmetrical in a and ß is precisely the same as the condition that the impulsive chain corresponding to a shall be reciprocal to ß. Following the analogy of our previous language we may describe two screw-chains so related as conjugate screw-chains of Inertia. 359. Harmonic screw-chains. We make one more application of the theory of screw-chains to the discussion of a kinetical problem. Let us suppose that we have any material system with n degrees of freedom in a position of stable equilibrium under the action of a conservative system of forces. If the system receive a small displacement, the forces will no longer equilibrate, but the system will be exposed to the action of a wrench on a screw-chain. We thus have two corresponding sets of screw-chains, one set being the chains about which the system is displaced, the other set for the wrenches which are evoked in consequence of the displacements. By similar reasoning to that which we have already used, it can be shown that these two corresponding chain systems are homographic. We can therefore find n screw-chains about which, if the system be displaced, a wrench will be evoked on the same screw-chain, and (the forces having a potential) it can be shown that this set of n screw-chains are co-reciprocal. If after displacement the system be released it will continue to make small oscillations. The nature of these oscillations can be completely exhibited by the screw-chains. To a chain a, regarded as an instantaneous screw-chain, will correspond the screw 0 as an impulsive screw-chain. To the chain a, regarded as the seat of a displacing twist, will correspond a wrench