A Treatise on the Theory of Screws

356] THE THEORY OF SCREW-CHAINS. 393 The first point to be noticed is, that the correspondence is unique. To the instantaneous chain a one impulsive screw-chain 0 will correspond. There could not be two screw-chains 0 and 0' which correspond to the same instan- taneous screw-chain a. For, suppose this were the ease, then the twist velocity imparted by the impulsive wrench on 0 could be neutralized by the impulsive wrench on 0'. We thus have the mass-chain remaining at rest in spite of the impulsive wrenches on 0 and 0'. These two wrenches must therefore neutralize, and as, by hypothesis, they are on different screw- chains, this can only be accomplished by the aid of the reactions of the constraints. We therefore find that 0 and 0' must compound into a wrench-chain which is neutralized by the reactions of the constraints. This is, however, impossible, for 0 and 0' can only compound into a wrench on a screw-chain of the original system, while all the reactions of the constraints form wrenches on the chains of the wholly distinct reciprocal system. We therefore see that to each instantaneous screw-chain a only one impulsive screw-chain 0 will correspond. It is still easier to show that to each impulsive screw-chain 0 only one instantaneous screw-chain a will correspond. Suppose that there were two screw-chains, a and a', either of which would correspond to an impulsive wrench on 0. We could then give the mass-chain, first, an impulsive wrench on 0 of intensity X, and make the mass-chain twist about a, and we could simultaneously give it an im- pulsive wrench on the same screw-chain 0 of intensity — X, and make the mass-chain twist about a'. The two impulses would neutralize, so that as a matter of fact the mass-chain received no impulse whatever, but the two twist velocities could not destroy, as they are on different screw-chains. We would thus have a twist velocity produced without any expenditure of energy. We have thus shown that in the n-system of screw-chains expressing the freedom of the mass-chain, one screw-chain, regarded as an instantaneous screw-chain, will correspond to one screw-chain, regarded as an impulsive screw-chain, and conversely, and therefore linear relations between the co-ordinates are immediately suggested. That there are such relations can be easily proved directly from the laws of motion (see Appendix, note 7). We therefore have established a case of screw-chain homography between the two systems, so that if 0x,...0n denote the co-ordinates of the impulsive screw-chain, and if a,, ... an denote the co-ordinates of the corresponding instantaneous screw-chain, we must have n equations of the type 01 = (ll)a1 + (12)a2 + (13)a3... +(1 n) an, 0n = (n 1) otj + (n 2) a2 +.........+ (nn) an>