A Treatise on the Theory of Screws

354] THE THEORY OF SCREW-CHAINS. 389 Let a and ß be two screw-chains, each consisting of /z screws, appro- priated one by one to the g elements of the mass-chain. If the system receive a twist about the screw-chain ß, while a wrench acts on the screw- chain a, some work will usually be lost or gained; if, however, no work be lost or gained, then the same will be true of a twist around a acting on a wrench on ß. In this case the screw-chains are said to be reciprocal. The relation, may be expressed somewhat differently, as follows:— If a mass-chain, only free to twist about the screw-chain a, be in equilibrium, notwithstanding the presence of a wrench on the screw-chain ß, then, conversely, a mass-chain only free to twist about the screw-chain ß will be in equilibrium, notwithstanding the presence of a wrench on the screw-chain a. This remarkable property of two screw-chains is very readily proved from the property of two reciprocal screws, of which property, indeed, it is only an extension. Let a, ... be the screws of one screw-chain, and ß, ... ßp those of the other. Let a/, of, ... a/ denote amplitudes of twists on «1; a2, &c., and let af, of, &c., denote the intensities of wrenches on a,, a.,, &c. Then, from the nature of the screw-chain, we must have = ai : «2" = a/ : a3", &c., Ä' ■ ßf = ßj -ß"^ ■ ßs', &c.; for as twists and wrenches are compounded by the same rules, the inter- mediate screws of the chain require that the ratio of two consecutive amplitudes of the twists about the chain shall coincide with the ratio of the intensities of the two corresponding wrenches. Denoting the virtual coefficient of ax and ßt by the symbol va,ß„ we have for the work done by a twist about a, against the screw-chain ß, 2a1'ß1"'sraiß1 + 2afßfis aA + &c., while for the work done by a twist about ß against thø screw-chain a we have the expression Zafßf^ß, + 2a2"/3i'«-a^3> &c. If the first of these expressions vanishes, then the second will vanish also. It will now be obvious that a great part of the Theory of Screws may be applied to the more general conceptions of screw-chains. The following theorem can be proved by the same argument used in the case when only a single pair of screws are involved. If a screw-chain 0 be reciprocal to two screw-chains a and ß, then 0 will be reciprocal to every screw-chain of the system obtained by compounding twists on a and ß.