188 THE THEORY OF SCREWS. [186- 



If we take three conjugate screws of inertia from the screw system as 

 screws of reference, then we have seen (97) that, if 1} 0.,, 3 , be the co 

 ordinates of a screw 0, we have 



where u lt u. 2 , u s are the values of u g with reference to the three conjugate 

 screws of inertia. 



Draw from any point lines parallel to 0, and to the three conj ugate screws 

 of inertia. If then a parallelepiped be constructed of which the diagonal is 

 the line parallel to 0, and of which the three lines parallel to the conjugate 

 screws are conterminous edges, and if r be the length of the diagonal, and 

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



x _a V a z a 

 r~ * IT = 2&amp;gt; r if* 



We see, therefore, that the parameter u appropriate to any screw is 

 inversely proportional to the parallel diameter of the ellipsoid 



u.?z&amp;gt; = H, 

 where H is a 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 system 

 of the third order, is proportional to the inverse square of the parallel 

 diameter of a certain ellipsoid, which may be called the ellipsoid of inertia ; 

 and a set of three conjugate diameters of the ellipsoid are parallel to a set 

 of three conjugate screws of inertia which belong to the screw system. 



We might also enunciate the property in the following 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 system, so 

 that its kinetic energy shall be constant. 



187. 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. W T e can then determine 

 three screws of the system parallel to these diameters ( 180), and these 

 three screws will be co-reciprocal, and also conjugate screws of inertia. 

 They will, therefore, ( 87), form what we have termed the principal screws 

 of inertia. When the screw system reduces to a pencil of screws of zero 

 pitch passing through a point, then the principal screws of inertia reduce 

 to the well-known principal axes. 



