196] FREEDOM OF THE THIRD ORDER. 193 



195. Wrench evoked by Displacement. 



By the aid of the quadric of the potential we shall be able to solve 

 the problem of the determination of the screw on which a wrench is evoked 

 by a twist about a screw 6 from a position of stable equilibrium. The 

 construction which will now be given will enable us to determine the screw 

 of the system on which the reduced wrench acts. 



Draw through the centre of the pitch quadric a line parallel to 6. Con 

 struct the diametral plane A of the quadric of the potential conjugate to 

 this line, and let X, p, be any two screws of the system parallel to a pair of 

 conjugate diameters of the quadric of the potential which lie in the plane 

 A. Then the required screw &amp;lt;/&amp;gt; is parallel to that diameter of the pitch 

 quadric which is conjugate to the plane A. 



For &amp;lt;f&amp;gt; will then be reciprocal to both X and //,; and as X, /A, 6 are 

 conjugate screws of the potential, it follows that a twist about 6 must evoke 

 a reduced wrench on &amp;lt;. 



196. Harmonic Screws. 



When a rigid body has freedom of the third order, it must have ( 106) 

 three harmonic screws, or screws which are conjugate screws of inertia, as 

 well as conjugate screws of the potential. We are now enabled to construct 

 these screws with facility, for they must be those screws of the screw system 

 which are parallel to the triad of conjugate diameters common to the ellipsoid 

 of inertia, and the quadric of the potential. 



We have thus a complete geometrical conception of the small oscillations 

 of a rigid body which has freedom of the third order. If the body be once 

 set twisting about one of the harmonic screws, it will continue to twist 

 thereon for ever, and in general its motion will be compounded of twisting 

 motions upon the three harmonic screws. 



If the displacement of the body from its position of equilibrium has 

 been effected by a small twist about a screw on the cylindroid which contains 

 two of the harmonic screws, then the twist can be decomposed into com 

 ponents on the harmonic screws, and the instantaneous screw about which 

 the body is twisting at any epoch will oscillate backwards and forwards 

 upon the cylindroid, from which it will never depart. 



If the periods of the twist oscillations on two of the harmonic screws 

 coincided, then every screw on the cylindroid which contains those harmonic 

 screws would also be a harmonic screw. 



If the periods of the three harmonic screws were equal, then every screw 

 of the system would be a harmonic screw. 



B. 13 



