A Treatise on the Theory of Screws

270 THE THEORY OF SCREWS. [254- The transformation having been effected, an important result is im- mediately deduced. Let the transformed function be denoted by then we have (11) ... + (66) a62, Ä=l(ll)a]; ft=^-(66)a6; Pe whence it appears that the six screws of reference are the common screws of the two systems. We thus find that in this special case of homography The six common screws of the two systems are co-reciprocal. The correspondence between impulsive screws and instantaneous screws is a particular case of the type here referred to. The six common screws of the two systems are therefore what we have called the principal screws of inertia, and they are co-reciprocal. 255. Correspondence of a Screw and a system. We shall sometimes have cases in which each screw of a system cor- responds not to a single screw but to a system of screws. For the sake of illustration, suppose the case of a quiescent rigid body with two degrees of freedom and let this receive an impulsive wrench on some screw situated anywhere in space. The movement which the body can accept is limited. It can, indeed, only twist about one of the singly infinite number of screws, which constitute a cylindroid. To any screw in space will correspond one screw on the cylindroid. But will it be correct to say, that to one screw on the cylindroid corresponds one screw in space ? The fact is, that there are a quadruply infinite number of screws, an impulsive wrench on any one of which will make the body choose the same screw on the cylindroid for its instantaneous movement. The relation of this quadruply infinite group is clearly exhibited in the present theory. It is shown in §128 that, given a screw a on the cylindroid, there is, in general one, but only one screw 0 on the cylindroid, an impulsive wrench on which will make the body commence to twist about a. It is further shown that any screw whatever which fulfils the single condition of being reciprocal to a single specified screw on the cylindroid possesses the same property. The screws corresponding to a thus form a five-system. The correspondence at present before us may therefore be enunciated in the following general manner. To one screw in space corresponds one screw on the cylindroid, and to one screw on the cylindroid corresponds a five-system in space.