578 EEPOET— 1887. 



had so puzzled the committee, reduced the body to rest, and then introduced the 

 subject 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. Here Mr. Anharmonic pointed out how 

 exactly modern geometry was adapted to supply the wants of Dynamics. 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 ? How do they 

 throw light on those elaborate oscillations which seem at present so inexplicable ? ' 

 ' This I shall explain,' said Anharmonic ; ' but 1 beg of you to give me your best 

 attention, for I think the theory of small oscillations will be found worthy of it. 



* Let us think of any screw a belonging to the system U, which expresses the 

 freedom of the body. If a be an instantaneous screw, there will of course be a 

 corresponding impulsive screw 6 also on U. If the body be displaced from a position 

 of equilibrium by a small twist about a, the uncompensated forces will produce a 

 wrench (p, which, without loss of generality, may also be supposed to he on U. 

 According as the screw a moves over U so will the two corresponding screws 

 6 and (}> also move over U. The system represented by a is homographic with both 

 the systems of 6 and of (p respectively. But two systems homographic with the 

 same system are homographic with each other. Accordingly, the system and the 

 (p system are homographic. There will therefore be a certain number of double 

 screws (not more than six) common to the systems 6 and (p. Each of these double 

 screws will of course have its correspondent in the a system, and we may call them 

 aj, Oj, &c., their number being equal to the degrees of freedom of the body. These 

 screws are most curiously related to the small oscillations. We shall first demon- 

 strate by experiment the remarkable property they possess.' 



The body was first brought to rest in its position of equilibrium. One of the 

 special screws a, haAang been carefully determined both in position and in pitch, 

 the body was displaced by a twist about this screw and was then released. As 

 the forces were uncompensated, the body of course commenced to move, but the 

 oscillations were of unparalleled simplicity. With the regularity of a pendulum 

 the body twisted to and fro on this screw, just as if it were actually constrained to 

 this motion alone. The committee were delighted to witness a vibration so graceful, 

 and, remembering the complex nature of the ordinary oscillations, they appealed to 

 Mr. Anharmonic for an explanation. This he gladly gave, not by means of com- 

 plex formulae, but by a line of reasoning that was highly commended by Mr. 

 Commonsense, and such that even Mr. Querulous could understand. 



' This pretty movement,' said Mr. Anharmonic, ' is due to the nature of the 

 screw Oj. Had I chosen any screw at random, the oscillations would, as we have 

 seen, be of a very complex type ; for the displacement will always evoke an uncom- 

 pensated wrench, in consequence of which the body will commence to move by 

 twisting about the instantaneous sei'ew corresponding to that wrench ; and of 

 course this instantaneous screw will usually be quite different from the screw about 

 which the displacement was made. But you will observe that a, has been chosen 

 as a screw in the instantaneous system, corresponding to one of the double screws 

 in the 6 and (p systems. When the body is twisted about a, a wrench is evoked 

 on the double screw, but as Oj is itself the instantaneous screw, corresponding to 



