APPENDIX II. 503 



encouragement from these discoveries ; they felt that they must be following the 

 right 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 corresponding 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 

 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 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, any 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 Orie-to-One, they called on him to explain how this special screw was to be 

 identified ; accordingly he proceeded : We have already ascertained that the 

 constraints permit 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 almost 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 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 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 impulsive 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 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 



