A Treatise on the Theory of Screws

APPENDIX II. 503 encouragement from these discoveries; they felt that they must be following the right path, and that the bounty of Nature had already bestowed on them some earnest of the rewards they were ultimately to receive. It was with eager anticipation that they now approached the great dynamical question. They were to see what would happen if the impulsive wrench were not neutralised by the reactions of the constraints. The body would then commence to move—that is, to twist about some screw which it would be natural to call the instantaneous screw. To trace the connection between the impulsive screw and the corresponding instantaneous screw was the question of the hour. Before the experiments were commenced, some shrewd member remarked that the issue had not yet been presented with the necessary precision. ‘ I understand,’ he said, ‘ that when you apply a certain impulsive wrench, the body will receive a definite twist velocity about a definite screw; but the converse problem is ambiguous. Unless the body be quite free, there are myriads of impulsive screws corresponding to but one instantaneous screw.’ The chairman perceived the difficulty, and not in vain did he appeal to the geometrical instinct of Mr One-to- One, who at once explained the philosophy of the matter, dissipated the fog, and disclosed a fresh beauty in the theory. ‘It is quite true,’ said Mr One-to-One, ‘that there are myriads of impulsive screws, any one of which may be regarded as the correspondent to a given instan- taneous screw, but it fortunately happens that among these myriads there is always one screw so specially circumstanced that we may select it as the correspondent, and then the ambiguity will have vanished.’ As several members were not endowed with the geometrical insight possessed by One-to-One, they called on him to explain how this special screw was to be identified; accordingly he proceeded:—‘We have already ascertained that the constraints permit the body to be twisted about any screw of the system, U. Out of the myriads of impulsive screws, corresponding to a single instantaneous screw, it almost always happens that one, but never more than one, lies on U. Thia is the special screw. No matter where the impulsive wrench may lie throughout all the realms of space, it may be exchanged for a precisely equivalent wrench lying on U. Without the sacrifice of a particle of generality, we have neatly circumscribed the problem. For one impulsive there is one instantaneous screw, and for one instantaneous screw there is one impulsive screw.’ The experiments were accordingly resumed. An impulsive screw was chosen, and its position and its pitch were both noted. An impulsive wrench was administered, the body commenced to twist, and the instantaneous screw was ascertained by the motion of marked points. The body was brought to rest. A new impulsive screw was then taken. The experiment was again and again repeated. The results were tabulated, so that for each impulsive screw the corresponding instantaneous screw was shown. Although these investigations were restricted to screws belonging to the system which expressed the freedom of the body, yet the committee became uneasy when they reflected that the screws of that system wei'e still infinite in number, and that consequently they had undertaken a task of infinite extent. Unless some