A Treatise on the Theory of Screws

K; ; - 388 THE THEORY OF SCREWS. [353, 353. Freedom of the eighth and higher orders. If the freedom be of the eighth order, then it is easily shown that the first screw of any other chain may be taken arbitrarily, and that even the second screw may be chosen arbitrarily from a three-system. Passing on to the twelfth order of freedom, the two first screws of the chain, as well as the amplitudes of their twists, may be chosen quite arbitrarily, and the rest of the chain is fixed. In the thirteenth order of freedom we can take the two first twists arbitrarily, while the third may be chosen anywhere on a cylindroid. It will not now be difficult to trace the progress of the chain to that unrestrained freedom it will enjoy when the mass-chain has 6/x degrees of freedom, when it is able to accept any displacement whatever. In the last stage, prior to that of absolute freedom, the system will have its position defined by 6/x — 1 co-ordinates. A screw-chain can then be chosen which is perfectly arbitrary in every respect, save that one of its screws must be reciprocal to a given screw. 354. Reciprocal Screw-Chains. We have hitherto been engaged with the discussion of the geometrical or kinematical relations of a mass-chain of y elements : we now proceed to the dynamical considerations which arise when the action of forces is considered. Each element of the mass-chain may be acted upon by one or more external forces, in addition to the internal forces which arise from the reaction of constraints. This group of forces must constitute a wrench appropriated to the particular element. For each element we thus have a certain wrench, and the entire action of the forces on the mass-chain is to be represented by a series of y wrenches. Recalling our definition of a screw-chain, it will be easy to assign a meaning to the expression, wrench on a screw-chain. By this we denote a series of wrenches on the screws of the chain, and the ratio of two consecutive intensities is given by the intermediate screw, as before. We thus have the general statement:— The action of any system of forces on a mass-chain may be represented by a wrench on a screw-chain. Two or more wrenches on screw-chains will compound into one wreuch on a screw-chain, and the laws of the composition are exactly the same as for the composition of twists, already discussed. Take, for example, any four wrenches on four screw-chains. Each set of four homologous screws will determine a four-system; the resulting wrench- chain will consist of a series of wrenches on these four-systems, each being the "parallel projection” of the other.