574 KEPOKT — 1887. 



occurred to tlie committee to try the effect of iutercliangino- the relation of these 

 screws. Thev accordingly arranged that the body should be left only free to twist 

 about Y, while a wrench was applied on X. Again the body did not stir. The 

 importance of this fact immediately arrested the attention of the more intelUgent 

 observers, for it established the following general law : If a wrench on X fails to 

 move a body only free to twist about Y, then a wrench on Y must be unable to 

 move a bodv only free to twist about X. It was determined to speak of two 

 screws when related in this manner as reciprocal. 



Some members of the committee did not at first realise the significance of this 

 discovery. Their difficulty arose from the restricted character of the experiments 

 by which the law of reciprocal screws had been suggested. They said, ' ^ou have 

 shown us that this law is observed in the case of a body only free to twist about 

 one screw at a time ; but how does this teach anything of the general case in 

 which the body is free to twist about whole shoals of screws? ' Mr. Commonsense 

 immediately showed that the discovery could be enunciated in a quite un- 

 objectionable form. ' The law of reciprocal screws,' he said, ' does not depend 

 upon the constraints or the limitations of the freedom. It may be expressed in this 

 .^-ay : — Ticu screivs are reciprocal 2chen a small twist about either can do no icork 

 against a wrench on the other. ^ 



This important step at once brought into view the whole geometry of the 

 reactions. Let us suppose that the freedom of the body was such that it could 

 twist about all the screws of a system which we shall call U . Let all the possible 

 reactions form wrenches on the screws of another system, V. It then appeared 

 that every screw upon U is reciprocal to every screw upon V. A body might 

 therefore be free to twist about every screw of V and still remain in equilibrium, 

 notwithstanding the presence of a wrench on every screw of U. A body free to 

 twist about all the screws of V can therefore be only partially free. Hence V 

 must be one of those few types of screw system already discussed. It was 

 accordingly found that the single screw, the cylindroid, and the set of hyper- 

 boloids completely described 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 

 rifht path, and that the bounty of Nature had already bestowed 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 

 neutralised by the reactions of the constraints. 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 connection between the impulsive screw and 

 the corresjiondiug instantaneous screw was the question of the hour. Before 

 the experiments were commenced, some shrewd member remarked that the issue 

 had not yet been presented with the necessary precision. ' I understand,' 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 

 ambio-uous. Unless the body be quite free, there are myriads of impulsive screws 

 corresponding to but one instantaneous screw.' The chairman ])erceived the 

 dilHculty, and not in vain did he appeal to the geometrical instinct of Mr. One-to- 

 One, who at once explained the philosophy of the matter, dissipated the fog, and 

 disclosed a fresh beauty in the theory. 



' It is quite true,' said Mr. One-to-One, ' that there are myriads of impulsive 

 screws, anj- one of which may be regarded as the correspondent to a given instan- 

 taneous 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 proceeded : — ' We have already ascertamed that the 

 constraints jiermit 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, 



