A Treatise on the Theory of Screws

390 THE THEORY OF SCREWS. [354, A screw-chain is defined by 6/z. — 1 data (§ 353). It follows that a finite number of screw-chains can be determined, which sball be reciprocal to 6/z -1 given screw-chains. It is, however, easy to prove that that number must be one. If two chains could be found to fulfil this condition, then every chain formed from the system by composition of two twists thereon would fulfil the same condition. Hence we have the important result— One screw-chain can always be determined which is reciprocal to 6/z -1 given screw-chains. Ihis is of course only the generalization of the fundamental proposition with respect to a single rigid body, that one screw can always be found which is reciprocal to five given screws (§ 25). 355. Twists on 6/x 4-1 screw-chains. Given 6/z +1 screw-chains, it is always possible to determine the ampli- tudes of certain twists about those chains, such that if those twists be applied in succession to a mass-chain of /z elements, the mass-chain shall, after the last twist, have resumed the same position which it had before the first. To prove this it is first necessary to show that from the system formed by composition of twists about two screw-chains, one screw-chain can always be found which is reciprocal to any given screw-chain. This is indeed the generalization of the statement that one screw can always be found on a cylindroid which is reciprocal to a given screw. The proof of the more general, theorem is equally easy. The number of screw-chains produced by composition of twists about the screw-chains a and ß is singly infinite. There can, therefore, be a finite number of screw-chains of this system reciprocal to a given screw-chain 3. But that number must be one; for if even two screw-chains of the system were reciprocal to 3, then every screw- chain of the system must also be reciprocal to 3. The solution of the original problem is then as follows Let a and ß be two of the given 6>+1 chains, and let 9 be the one screw-chain which is reciprocal to the remaining 6/z - 1 chains. Since the 6/z + 1 twists are to neutralize, the total quantity of work done against any wrench-chain must be zero. Take, then, any wrench-chain on 3. Since this is reciprocal to 6/z - I of the screw-chains, the twists about these screw-chains can do no work against a twist on 9 It follows that the amplitudes of the twists about a and ß must be such that the total amount of work done must be zero. For this to be the case, the two twists on a and ß must compound into one twist on the screw-chain v, which belongs to the system (aß), and is also reciprocal to 3. This defines the ratio of the amplitudes of the two twists on a and ß. We may in fact draw any cylmdroid containing three homologous screws of a, ß, and y then the ratio of the sines of the angles into which y divides tbe angle between a and ß is the ratio of the amplitudes of the twists on a and ß. In a