TRANSACTIONS OF SECTION A. 577 



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 has a monotonous 

 zero of pitch. Our general conceptions of mobility have thus been horribly 

 mutilated and disfigured before they 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 problem the wi-enches 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 theoi-y 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 summarised the results. He said that a geometrical 

 solution had been obtained of every conceivable problem as to the efiect of 

 impulse on a rigid body. The impulsive screws and the corresponding instanta- 

 neous 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 problem 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 neutralised by the reaction of the constraints. 

 Let the body be displaced fi"om its initial position by a small twist. The wrench 

 will no longer be neutralised by the reaction of the constraints ; accordingly when 

 the body is released it will commence to move. So far as the present investiga- 

 tions are concerned these movements are small oscillations. Attention was there- 

 fore 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, un- 

 fortunately, this screw is not fixed, it is indeed moving about in such an embarrass- 

 ing 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 oscUlations 

 in such an unsatisfactory state. Success had hitherto guided their eflx)rts. 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 any 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 tlie 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 how geometry could viviiy the theory. 

 Fortunately this view prevailed, and new experiments were commenced under the 

 direction of Mr. Anharmonic, who first quelled the elaborate oscillations which 



1887. p p 



