A Treatise on the Theory of Screws

502 THE THEORY OF SCREWS. gations if these reactions could be first studied and their number and position ascertained. To this suggestion Mr Cartesian demurred. He urged that it would involve an endless task. ‘ Look,’ he said, 1 at the complexity of the constraints : how the body rests on these surfaces here; how it is fastened by links to those points there; how there are a thousand-and-one ways in which reactions might originate.’ Mr C ommonsense and other members of the committee were not so easily deterred, and they determined to work out the subject thoroughly. At first they did not see their way clearly, and much time was spent in misdirected attempts. At length they were rewarded by a curious and unexpected discovery, which suddenly rendered the obscure reactions perfectly transparent. A trial was being made upon a body which had only one degree of freedom ; was, in fact, only able to twist about a single screw, Z. Another screw, Z, was speedily found, such that a wrench thereon failed to disturb the body. It now occurred to the committee to try the effect of interchanging the relation of these screws. They accordingly arranged that the body should be left only free to twist about Y, while a wrench was applied on X Again the body did not stir. The importance of this fact immediately arrested the attention of the more intelligent observers, for it established the following general law : If a wrench on X fails to move a body only free to twist about Y, then a wrench on Y must be unable to move a body only free to twist about X. It was determined to speak of two screws when related in this manner as reciprocal. Some members of the committee did not at first realise the significance of this discovery. Their difficulty arose from the restricted character of the experiments by which the law of reciprocal screws had been suggested. They said, ‘ You have shown us that this law is observed in the case of a body only free to twist about one screw at a time; but how does this teach anything of the general case in which the body is free to twist about whole shoals of screws V Mr Commonsense immediately showed that the discovery could be enunciated in a quite un- objectionable form. ‘The law of reciprocal screws,’ he said, ‘does not depend upon the constraints or the limitations of the freedom. It may be expressed in this way -.—Two screws are reciprocal when a small twist about either can do no work against a wrench on the other.' This important step at once brought into view the whole geometry of the reactions. Let us suppose that the freedom of the body was such that it could twist about all the screws of a system which we shall call U. Let all the possible reactions form wrenches on the screws of another system, V. It then appeared that every screw upon U is reciprocal to every screw upon V. A body might therefore be free to twist about every screw of V and still remain in equilibrium, notwithstanding the presence of a wrench on every screw of U. A body free to twist about all the screws of Y can therefore be only partially free. Hence V must be one of those few types of screw system already discussed. It was accordingly found that the single screw, the cylindroid, and the set of hyper- boloids completely described every conceivable reaction from the constraints just as they described every conceivable kind of freedom. The committee derived much