TKAXSACTIONS OF SECTION A. 575 



it always happens that one, but never more than one, lies on U. This is the special 

 screw. No matter where the impulsive wrench may lie throughout all the realms 

 of space, it may always 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 irnpulsive 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 impidsive screw was then taken. The experiment was again and ao-ain 

 repeated. The results were tabidated, 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 were still infinite in number, 

 and that consequently they had undertaken a task of infinite extent. Unless some 

 compendious law should be discovered, which connected the impulsive screw 

 with the instantaneous screw, their experiments would indeed be endless. Was 

 it likely that such a law could be found — was it even likely that such a law 

 existed ? Mr. Querulous decidedly tliought not. He pointed out how the body 

 was of the most hopelessly irregular shape and mass, and how the constramts 

 were notoriously of the most embarrassing description. It was, therefore he 

 thought, idle to search for any geometrical law connecting the impulsive screw 

 and the instantaneous screw. He moved that the whole inquiry be abandoned. 

 These sentiments seemed to be shared by other members of the committee. Even 

 the resolution of the chairman began to quail before a task of infinite magnitude. 

 A crisis was imminent — when Mr. Anharmonic rose. 



' Mr. Chairman,' he said, ' Geometry is ever ready to help even the most 

 humble inquirer into the laws of nature, but Geometry reserves her most gracious 

 gifts for those who interrogate Nature in the noblest and most comprehensive spirit. 

 That spirit has been ours during this research, and accordingly Geometry in this 

 our emergency places her choicest treasures at our disposal. Foremost amono- these 

 is the powerful theory of homographic systems. By a few bold extensions we 

 create a comprehensive theory of homographic screws. All the impulsive screws 

 form one system, and all the instantaneous screws form another system, and 

 these two systems are homographic. Once you have realised this, you will find 

 your present difticulty cleared away. You will only have to determine a few pairs 

 of impulsive and instantaneous screws by experiment. The number of such pairs 

 need never be more than seven. When these have been found, the homography is 

 completely known. The instantaneous screw corresponding to every fmpulsive 

 screw wHl then be completely determined by geometry both pure and beautiful.' 

 To the delight and amazement of the committee, Mr. Anharmonic demonstrated 

 the truth of his theory by the supreme test of fulfilled prediction. When the 

 observations had provided him with a number of pairs of screws, one more than 

 the number of degrees of freedom of the body, he was able to predict with in- 

 fallible accuracy the instantaneous screw corresponding to any impulsive screw. 

 Chaos had gone. Sweet order had come. 



A few days later the chairman summoned a special meeting in order to hear 

 from Mr. Anharmonic an account of a discoverj- he had just made, which he 

 believed to be of signal importance, and which he' was anxious to demonstrate by 

 actual experiment. Accordingly the committee assembled, and the geometer pro- 

 ceeded as follows : — 



'You are aware that two_ homographic ranges on the same ray possess two 

 double points, whereof each coincides with its correspondent ; more generally when 

 each pomt in space, regarded as belonging to one homographic system, has its 

 correspondent belonging to another system, then there are four cases in which a 

 point coincides with its correspondent. These are known as the four double points, 

 and they possess much geometrical interest. Let us now create conceptions of an 



