DYNAMICS OF A RIGID BODY. 141 



the ellipsoid 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 given screw 6 of 

 the complex. The construction which will now be given 

 will enable us to determine the screw of the complex on 

 which the reduced wrench acts. 



Draw through the centre of the pitch quadric a 

 line parallel to 9. Construct the diametral plane A of 

 the ellipsoid of the potential conjugate to this line, and 

 let X, fj. be any two screws of the complex parallel to a 

 pair of conjugate diameters of the ellipsoid of the poten- 

 tial which lie in the plane A. Then the required screw 

 is parallel to that diameter of the pitch quadric which 

 is conjugate to the plane A. 



For will then be reciprocal to both X and /* ; and 

 as X, //, 9 are conjugate screws of the potential, it fol- 

 lows that a twist about 9 must evoke a reduced wrench 

 on $. 



129. Harmonic Screws. When a rigid body has free 

 dom of the third order, it must have ( 74) three harmonic 

 screws, or screws which are conjugate screws of inertia L 

 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 complex which 

 are parallel to the triad of common conjugate diameters 

 of the ellipsoid of inertia, and the ellipsoid of the poten- 

 tial. 



We have thus a complete geometrical conception of 

 the small oscillations of a rigid body which has free- 

 dom 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 har- 

 monic screws. 



If the displacement of the body from its position of 



