A Treatise on the Theory of Screws

506 THE THEORY OF SCREWS. generality you prize so much and put the theory into some familiar shape that ordinary mortals can understand.’ Mr Anharmonic would not condescend to comply with this request, so the chairman called upon Mr One-to-One, who somewhat ungraciously consented. ‘I feel, said he, ‘the request to be an irritating one. Extreme cases frequently make bad illustrations of a general theory. That zero multiplied by infinity may be anything is not surely a felicitous exhibition of the perfections of the mul- tiplication table. It is with reluctance that I divest the theory of its flowing geometrical habit, and present it only as a stiff conventional guy from which true grace has departed. ‘ Let us suppose that the rigid body, instead of being constrained as heretofore in a perfectly general manner, is subjected merely to a special type of constraint. Let it in fact be only free to rotate around a fixed point. The beautiful fabric of screws, which so elegantly expressed the latitude permitted to the body before, has now degenerated into a mere horde of lines all stuck through the point. Those varieties in the pitches of the screws which gave colour and richness to the fabric have also vanished, and the pencil of degenerate screws have a monotonous zero of pitch. Our general conceptions of mobility have thus been horribly mutilated and disfigured before they can be adapted to the old and respectable problem of the rotation of a rigid body about a fixed point. For the dynamics of this problem the wrenches assume an extreme and even monstrous type. Wrenches they still are, as wrenches they ever must be, but they are wrenches on screws of infinite pitch; they have even ceased to possess definite screws as homes of their own. We often call them couples. ‘Yet so comprehensive is the doctrine of the principal screws of inertia that even to this extreme problem the theory may be applied. The principal screws of inertia reduce in this special case to the three principal axes drawn through the point. In fact we see that the famous property of the principal axes of a rigid body is merely a very special application of the general theory of the principal screws of inertia. Every one who has a particle of mathematical taste lingers with fondness over the theory of the principal axes. Learn therefore,’ says One-to-One in conclusion, ‘how great must be the beauty of a doctrine which comprehends the theory of principal axes as the merest outlying detail.’ Another definite stage in the labours of the committee had now been I’eaclied, and accordingly the chairman summarised the results. He said that a geometrical solution had been obtained of every conceivable problem as to the effect of impulse on a rigid body. The impulsive screws and the corresponding instan- taneous screws formed two homographic systems. Each screw in one system determined its corresponding screw in the other system, just as in two anharmonic ranges each point in one determines its correspondent in the other. The double screws of the two homographic systems are the principal screws of inertia. He remarked in conclusion that the geometrical theory of homography and the present dynamical theory mutually illustrated and interpreted each other. There was still one more problem which had to be brought into shape by geometi-y and submitted to the test of experiment.