134 DYNAMICS OF A RIGID BODY. 



Draw from any point lines parallel to 0, and to- 

 the three conjugate screws of inertia. If then a pa- 

 rallelepiped be constructed of which the diagonal 

 is the line parallel to 0, and of which the three 

 lines parallel to the conjugate screws are contermi- 

 nous edges, and if r be the length of the diagonal, and 

 x y y, z the lengths of the edges, then we have 



x fi y fi z a 

 r =i 'r = ' r = 3 * 



We see, therefore! that the parameter u appropriate 

 to any screw 9 is inversely proportional to the parallel 

 diameter of the ellipsoid 



where His & certain constant. 



Hence we have the following theorem : The kinetic 

 energy of a rigid body, when twisting with a given twist 

 velocity about any screw of a complex of the third order, 

 is proportional to the inverse square of the parallel dia- 

 meter of a certain ellipsoid, which may be called the 

 ellipsoid of inertia ; and a set of three conjugate diame- 

 ters of the ellipsoid are parallel to a set of three conjugate 

 screws of inertia which belong to the screw complex. 



We might also enunciate the property in the follow- 

 ing manner : Any diameter of the ellipsoid of inertia is 

 proportional to the twist velocity with which the body 

 should twist about the parallel screw of the screw com- 

 plex, so that its kinetic energy shall be constant. 



1 20. The Principal Screws of Inertia. It will simplify 

 matters to consider that the ellipsoid of inertia is con- 

 centric with the pitch quadric. It will then be possible 

 to find a triad of common conjugate diameters to the 

 two ellipsoids. We can then determine three screws 

 of the complex parallel to these diameters ( H5)> 



