A Treatise on the Theory of Screws

368 THE THEORY OF SCREWS. [338- imposed by the constraints. The displacement of each element could, however, have been effected by a twist of appropriate amplitude about a screw specially correlated to that element. The total effect of the displace- ment could, therefore, have been produced by giving each element a certain twist about a certain screw. 339. The Graphic and Metric Elements. In the lowest type of freedom which the mass-chain can possess (short of absolute fixity) the freedom is of the first order, and any position of the mass-chain admits of specification by a single co-ordinate. In such a case the screw appropriate to each element is unique, and is completely determined by the constraints both in position and in pitch. The ratio of the amplitude of each twist to the amplitudes of all the other twists is also prescribed by the constraints. The one co-ordinate which is arbitrary may be conveniently taken to be the amplitude of the twist about the first screw. To each value of this co-ordinate will correspond a possible position of the mass-chain. As the ratios of the amplitudes are all known, and as the first amplitude is given, then all the other amplitudes are known, and consequently the position assumed by every element of the mass-chain is known. The whole series of screws and the ratios of the amplitudes thus embody a complete description of the particular route along which the mass-chain admits of displacement. The actual position of the mass-chain is found by adding to the purely graphic element which describes the route a metric element, to indicate the amplitude through which the mass-chain has travelled along that route. This amplitude is the arbitrary co-ordinate. 340. The Intermediate Screw. It will greatly facilitate our further progress to introduce a conventional process, which will clearly exhibit the determinate character of the ratios of the amplitudes in the screw series. Consider the two first screws, ax and a2 of the series. Draw the cylindroid (a,, a2) which contains these two screws. Since ctj and a2 are appropriated to two different elements of the mass-chain, no kinematical significance can be attached to the composition of the two twists on a, and a2. If, however, the two twists on «1 and a2, having the proper ratio of amplitudes, had been applied to a single rigid body, the dis- placement produced is one which could have been effected by a single twist about a single screw on the cylindroid (a1: a2). If this inter- mediate screw be given, the ratio of the amplitudes of the twists on the given screws is determined. It is in fact equal to the ratio of the sines of the angles into which the intermediate screw divides the angle between the two given screws. With a similar significance we may conceive an intermediate screw inserted between every consecutive pair of the p original screws.