428 



NA 7 URE 



{Sept. I, 1887 



'invincible attachment tothex,ji', s, which he regarded as the ne 

 plus ultra of dynamics. " Why will you burden the science," he 

 sighs, "with all these additional names? Can you not express 

 What you want without talking about cylindroids, and twists, and 

 wrenches, and impulsive screws, and instantaneous screws, and 

 all the rest of it ? " "No," said Mr. One-to-One, "there can 

 be no simpler way of stating the results than that natural 

 method we have followed. You would not object to the lan- 

 guage if your ideas of the natural phenomena had been suffi- 

 ciently capacious. We are dealing with questions of perfect 

 generality, and it would involve a sacrifice of generality were we 

 to speak of the movement of a body except as a twist, or of a 

 system of forces except as a wrench." 



" But," said Mr. Commonsense, "can you not as a conces- 

 sion to our ignorance tell us something in ordinary language 

 which will give an idea of what you mean when you talk of your 

 'principal screws of inertia?' Pray for once sacrifice this 

 generality you prize so much and put the theory into some 

 extreme shape that ordinary mortals can understand." 



Mr. Anharmonic would not condescend to comply with this 

 request, so the chairmrn called upon Mr. One-to-One, who 

 somewhat ungraciously consented. "I feel," said he, "the 

 •request to be an irritating one. Extreme cases generally make 

 bad illustrations of a general theory. That zero multiplied by 

 infinity may be anything is surely not a felicitous exhibition of 

 the perfections of the multiplication table. It is with reluctance 

 that I divest the theory of its flowing geometrical habit, and 

 present it only as a stiff conventional guy from which true grace 

 has departed. 



" Let us suppo<;e that the rigid body, instead of being con- 

 strained as heretofore in a perfectly general manner, is subjected 

 merely to a special type of constraint. Let it, in fact, be only 

 free to rotate around a fixed point. The beautiful fabric of 

 -screws, which so elegantly expressed the latitude permitted to 

 the body before, has now degenerated into a mere horde of lines 

 all stuck through the point. Those varieties in the pitches of 

 the screws which gave colour and richness to the fabric have 

 also vanished, and the pencil of degenerate screws have a mono- 

 tonous zero of pitch. Our general c inceptions of mobility have 

 thus been horribly mutilated and disfigured before thsy can be 

 adapted to the old and respectable problem of the rotation of a 

 rigid body about a fixed point. For the dynamics of this prob- 

 lem the wrenches assume an extreme and even monstrous type. 

 Wrenches they still are, as wrenches they ever must be, but they 

 are wrenches on screws of infinite pitch ; they have ceased to 

 possess definite screws as homes of their own. We often call 

 ■ them couples. 



"Yet so comprehensive is the doctrine of the principal screws 

 of inertia that even to this extreme problem the theory may be 

 applied. The principal screws of inertia reduce in this special 

 • case to the three principal axes drawn through the point. In 

 fact, we see that the famous property of the principal axes of a 

 rigid body is merely a very special application of the general 

 theory of the principal screws of inertia. Everyone who has a 

 particle of mathematical taste lingers with fondness over the 

 theory of the principal axes. Learn, therefore, " says One-to-One 

 in conclusion, " how great must be the beauty of a doctrine which 

 comprehends the theory of principal axes as the merest outlying 

 detail." 



Another definite stage in the labours of the committee had 

 now been reached, and accordingly the chairman summarized 

 the results. He said that a geometrical solution had been ob- 

 tained of every conceivable problen as to the effect of impulse 

 •on a rigid body. The impulsive screws and the corresponding 

 instantaneous screws formed two homographic systems. Each 

 screw in one system determined its corresponding screw in the 

 other system, just as in two anharmonic ranges each point in one 

 determines its correspondent in the other. The double screws of 

 the two homographic systems are the principal screws of inertia. 

 He remarked, in conclusion, that the geometrical theory of 

 homography and the present dynamical theory mutually illustrated 

 and interpreted each other. 



There was still one more problen which had to be brought 

 into shape by geometry, and submitted to the test of experiment. 



The body is lying at rest though gravity and many other 

 forces are acting upon it. These forces constitute a wrench 

 which must lie upon a screw of the reciprocal system, inasmuch 

 : as it is neutralized by the reaction of the constraints. Let the 

 body be displaced from its initial position by a small twist. The 

 wrench will no longer be neutralized by the reaction of the con- 



straints ; accordingly when the body is released it will commence 

 to move. So far as the present investigations are concerned 

 these movements are small oscillations. Attention was therefore 

 directed to these small oscillations. The usual observations were 

 made, and Helix reported them to be of a very perplexing kind. 

 " Surely," said the chairman, " you find the body twisting about 

 some screw, do you not?" "Undoubtedly," said Helix ; "the 

 body can only move by twisting about some screw ; but, unfor- 

 tunately, this screw is not fixed, it is indeed moving about in such 

 an embarrassing manner that I can give no intelligible account 

 of the matter." The chairman appealed to the committee not to 

 leave the interesting subject of small oscillations in such an un- 

 satisfactory state. Success had hitherto guided their efforts. Let 

 them not separate without throwing the light of geometry on 

 this obscure subject. 



Mr. Querulous here said he must be heard. He protested 

 against further waste of time ; there was nothing for them to do. 

 Everybody knew how to investigate small oscillations ; the 

 equations were given in every book on mechanics. You had 

 only to write down these equations, and scribble away till you 

 got out something or other. But the more intelligent members 

 of the committee took the same view as the chairman. They 

 did not question the truth of the formulae which to Querulous 

 seemed all-sufficient, but they wished to see what geometry could 

 do for the subject. Fortunately this view prevailed, and new 

 experiments were commenced under the direction of Mr. An- 

 harmonic. He first quelled the elaborate oscillations which had 

 so puzzled the committee ; he reduced the body to rest, and then 

 introduced the subject as follows : — 



" The body now lies at rest. I displace it a little, and I 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 

 neutralized 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 connexion between the two systems of 

 screws. Here Mr. Anharmonic pointed out how exactly modem 

 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 statement was put to the 

 test. Again and again the body was di'jplaced 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 oscilla- 

 tions which seem at present so inexplicable ? " 



"This I shall explain," said Anharmonic ; "but I 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, Q, also on U. If the body be displaced from a 

 position of equilibrium by a small twist about a, then the un- 

 compensated forces produce a wrench, ^, which, without loss of 

 generality, may also be supposed to lie on U. According as the 

 screw a moves over U so will the two corresponding screws fl 

 and (^ also move over U. The system represented by o is 

 homographic with both the systems of B and of ^ respectively. 

 But two systems homographic with the same system are homo- 

 graphic with each other. Accordingly the Q system and the ^ 

 system are homographic. There will therefore be a certain 

 number of double screws (not more than six) common to the 

 systems and (p. Each of these double screws will of course 

 have its correspondent in the o system, and we may call them 

 a,, a.,, &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 deinonstrate by experin 

 the remarkable property they possess." 



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

 One of the special screws a having been carefully determined 



o uie. 



d 



