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 formulae 

 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 longer 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, are not these purely statical results 1 How do they 



