Sept. I, 18S7J 



NA TURE 



427 



t about all the screw s of V can therefore be only partially 



Hence V must be one of those few types of screw system 



idy discussed. It was, accordingly, found that the single 



w, the cylindroid, and the set of hyperboloids completely 



Lirsciibed every conceivable reaction from the constraints just as 



they described every conceivable kind of freedom. The committee 



derived much encouragement from these discoveries ; they felt 



that they must be following the right path, and that the bounty 



of Nature had already iiestowed 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 neutralized by the reactions of 

 the con--traints. 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 connexion between 

 the impulsive screw and the corresponding instantaneous screw 

 was the question of the hour. Before the experiments were com- 

 menced, some shrewd member remarked that the issue had not 

 yet been presented with the necessary precision. "I under- 

 stand, " 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 

 geome:rical instinct of Mr. One-to-One, who at once explained 

 the philosophy of the matter, dissipated the fug, and disclo.';ed 

 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 instantaneous 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 pro- 

 ceeded : — " We have already ascertained that the constraints per- 

 mit 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 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 

 impulsive screw 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 bdy was brought to rest. A new im- 

 pulsive 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 be- 

 longing 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 thought not. He pointed 

 out how the body was of the most hopelessly irregular shape 

 and mass, and how the constraints were notoriously of the most 

 embarrassing description. It was therefore, he thought, idle to 

 search for any geometiical 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 thoie who interro- 



gate Nature in the noblest and most comprehensive spirit. That 

 spirit has been ours duiing this research, and accordingly Geo- 

 metry in this our emergency places her choicest treasures at our 

 disposal. Foremost among these is the powerful theory of 

 homographic systems. By a few bold extensions we create a 

 comprehensive theory of homographic screws. All the impul- 

 sive screws foim one system, and all the instantaneous screws 

 form another system, and these two systems are homographic. 

 Once you have realized this, you will find your present difficulty 

 cleared away. You will only have to determine a few pairs of 

 impulsive and instantaneous screws by experiment. The num- 

 ber 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 impulsive screw will 

 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 infallible 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 discovery 

 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 point in space, re- 

 garded as belonging to one homographic system, has its corre- 

 spondent 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 geo- 

 metrical interest. Let us now create conceptions of an analo- 

 gous character suitably enlarged for our present purpose. We 

 have discovered that the impulsive screws and the corresponding 

 instantaneous screws form two homographic systems. There 

 will be a certain limited number (never more than six) of double 

 screws common to these two systems. As the double points in 

 the homography of point systems are fruitful in geometry, so the 

 double screws in the homography of Ecrew systems are fruitful in 

 dynamics." 



A question for experimental inquiry could now be distinctly 

 stated. Does a double screw possess the property that an im- 

 puUive wrench delivered thereon will make the body commence 

 to move by twisting about the same screw ? This was imme- 

 diately tested. Mr. Anharmonic, guided by the indications of 

 homography, soon pointed out the few double screws. One of 

 these was chosen ; a vigorous impulsive wrench was imparted there- 

 on. The observations were conducted as before : the anticipated 

 result was triumphantly verified, for the body commenced to twist 

 about the identical screw on which the wrench was imparted. 

 The other double screws were similarly tried, and with a like 

 result. In each case the instantaneous screw was identical both 

 in pitch and in position with the impulsive screw. 



" But surely," said Mr. Querulous, " there is nothing wonder- 

 ful in this. Who is surprised to learn that the body twists about 

 the same screw as that on which the wrench was administered ? 

 I am sure I could find many such screws. Indeed, the real 

 wonder is not that the impulsive screw and the instantaneous 

 screw are ever the same, but that they are ever different." 



And Mr. Querulous proceeded to illustrate his views by ex- 

 periments on the rigid body. He gave the body all sorts of 

 impulses, but, in spite of all his endeavours, the body invariably 

 commenced to twist about some ; crew which was not the impul- 

 sive screw. "You may try till Doomsday," said Mr. Anhar- 

 monic, "you will never find any besides the few I have 

 indicated." 



It was thought convenient to assign a name to these remark- 

 able screws, and they were accordingly designated \^^ principal 

 screws of ificrtia. There are, for example, six principal screws 

 of inertia when the body is perfectly free, and two when the 

 body is free to twist about the screws of a cylindroid. The 

 committee regarded the discovery of the principal screws of 

 inertia as the most remarkable result they had yet obtained. 



Mr. Cartesian was very unhappy. The generality of the 

 subject was too great for his comprehension. He had an 



