508 THE THEORY OF SCREWS. 



throw light on those elaborate oscillations 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 also on U. If the body be displaced from a position 

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

 wrench &amp;lt;j&amp;gt;, 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 

 $ and &amp;lt;f&amp;gt; also move over U. The system represented by a is homographic with both 

 the systems of and of &amp;lt; respectively. But two systems homographic with the 

 same system are homographic with each other. Accordingly, the 6 system and the 

 &amp;lt; system are homographic. There will therefore be a certain number of double 

 screws (not more than six) common to the systems 6 and &amp;lt;f&amp;gt;. Each of these double 

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

 a 1} o. 2 , &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 having 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 to which even Mr Querulous urged no objection. 



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

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

 seen, be of a very complex type ; for the displacement will evoke an uncompensated 

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

 about the instantaneous screw 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 t has been chosen as a screw 

 in the instantaneous system, corresponding to one of the double screws in the 6 and 

 &amp;lt;f&amp;gt; systems. When the body is twisted about c^ a wrench is evoked on the double 

 screw, but as a, is itself the instantaneous screw, corresponding to that double 

 screw, the only effect of the wrench will be to make the body twist about 04. 

 Thus we see that the body will twist to and fro on c^ for ever. Finally, we can 

 show that the most elaborate oscillations the body can possibly have may be 

 produced by compounding the simple vibrations on these screws c^, a. 2 , &amp;lt;fec. 



Great enlightenment was thus diffused over the committee, and now Mr 

 Querulous began to think there must be something in it. Cordial unani- 



