A Treatise on the Theory of Screws

508 THE THEORY OF SCREWS. throw light on those elaborate oscillations which seem at present so inexplicable 1 ’ ‘This I shall explain,’ said Anharmonic ; ‘but I beg of you to give me youi1 best attention, for I think the theory of small oscillations will be found worthy of it. ‘ Let us think of any screw a belonging to the system 17, which expresses the freedom of the body. If a be an instantaneous screw, there will of course be a corresponding impulsive screw 0 also on U. If the body be displaced from a position of equilibrium by a small twist about a, then the uncompensated forces produce a wrench </>, which, without loss of generality, may also be supposed to lie on U. According as the screw a moves over U so will the two corresponding screws 0 and </> also move over U. The system represented by a is homographic with both the systems of 0 and of </> respectively. But two systems homographic with the same system are homographic with each other. Accordingly, the 0 system and the system are homographic. There will therefore be a certain number of double screws (not more than six) common to the systems 0 and </>. Each of these double screws will of course have its correspondent in the a system, and we may call them a,, a.,, &c., their number being equal to the degrees of freedom of the body. These screws are most curiously related to the small oscillations. We shall first demon- strate by experiment the remarkable property they possess.’ The body was first brought to rest in its position of equilibrium. One of the special screws a having been carefully determined both in position and in pitch, the body was displaced by a twist about this screw and was then released. As the forces were uncompensated, the body of course commenced to move, but the oscillations were of unparalleled simplicity. With the regularity of a pendulum the body twisted to and fro on this screw, just as if it were actually constrained to this motion alone. The committee were delighted to witness a vibration so graceful, and, remembering the complex nature of the ordinary oscillations, they appealed to Mr Anharmonic for an explanation. This he gladly gave, not by means of com- plex formulæ, but by a line of reasoning that was highly commended by Mr C'oininonsense, and to which even Mr Querulous urged no objection. ‘This pretty movement,’ said Mr Anharmonic, ‘is due to the nature of the screw oj. Had I chosen any screw at random, the oscillations would, as we have seen, be of a very complex type; for the displacement will evoke an uncompensated wrench, in consequence of which the body will commence to move by twisting about the instantaneous screw corresponding to that wrench ; and of course this instantaneous screw will usually be quite different from the screw about which the displacement was made. But you will observe that a, has been chosen as a screw in the instantaneous system, corresponding to one of the double screws in the 0 and ö systems. When the body is twisted about a2 a wrench is evoked on the double screw, but as a] is itself the instantaneous screw, corresponding to that double screw, the only effect of the wrench will be to make the body twist about a1. Thus we see that the body will twist to and fro on for ever. Finally, we can show that the most elaborate oscillations the body can possibly have may be produced by compounding the simple vibrations on these screws ap a2, &c.’ Great enlightenment was thus diffused over the committee, and now Mr Querulous began to think there must be something in it. Cordial unani-