A Treatise on the Theory of Screws

 APPENDIX II. 507 The body is lying at rest though gravity and many other forces are acting upon it. These forces constitute a wrench which must lie upon a screw of the reciprocal system, inasmuch as it is neutralised by the reaction of the constraints. Let the body be displaced from its initial position by a small twist. The wrench will no longer be neutralised by the reaction of the constraints; accordingly when the body is released it will commence to move. So far as the present investiga- tions are concerned these movements are small oscillations. Attention was there- fore directed to these small oscillations. The usual observations were made, and Helix reported them to be of a very perplexing kind. ‘Surely,’ said the chairman, ‘you find the body twisting about some screw, do you not?’ ‘Undoubtedly,’ said Helix; ‘the body can only move by twisting about some screw; but, un- fortunately, this screw is not fixed, it is indeed moving about in such an embarrass- ing manner that I can give no intelligible account of the matter.’ The chairman appealed to the committee not to leave the interesting subject of small oscillations in such an unsatisfactory state. Success had hitherto guided their efforts. Let them not separate without throwing the light of geometry on this obscure subject. Mr Querulous here said he must be heard. He protested against any further waste of time; it was absurd. Everybody knew how to investigate small oscil- lations ; the equations were given in every book on mechanics. You had only to write down these equations, solve, these equations again for the thousandth time and the thing was done. But the more intelligent members of the committee took the same view as the chairman. They did not question the truth of the formulæ which to Querulous seemed all sufficient, but they wished to see whether geometry could not illuminate the subject. Fortunately this view prevailed, and new ex- periments were commenced under the direction of Mr Anharmonic, who first quelled the elaborate oscillations which had so puzzled the committee, reduced the body to rest, and then introduced the discussion as follows :— ‘ The body now lies at rest. I displace it a little, and hold it in its new position. The wrench, which is the resultant of all the varied forces acting on the body, is no longei- completely neutralised by the reactions of the constraints. Indeed, I can feel it in action. Our apparatus will enable us to measure the intensity of this wrench, and to determine the screw on which it acts.’ A series of experiments was then made, in which the body was displaced by a twist about a screw, which was duly noted, while the corresponding evoked wrench was determined. The pairs of screws so related were carefully tabulated. When we remember the infinite complexity of the forces, of the constraints and of the constitution of the body, it might seem an endless task to determine the connection between the two systems of screws. Mr Anharmonic pointed out how modern geometry supplied the wants of Dynamics. As in the previous case the two screw systems were homographic, and when a number of pairs, one more than the degrees of freedom of the body, had been found all was determined. This state- ment was put to the test. Again and again the body was displaced in some new fashion, but again and again did Mr Anharmonic predict the precise wrench which would be required to maintain the body in its new position. ‘ But,’ said the chairman, 1 are not these purely statical results 1 How do they