156 DYNAMICS OF A RIGID BODY. 



pulsive screw can be determined when the rigid body 

 is perfectly free. It will be observed that the connexion 

 between the two screws depends only upon the three 

 principal axes through the centre of inertia, and the 

 radii of gyration about these axes. We may express 

 this result more compactly by the familiar concep- 

 tion of the momental ellipsoid. The centre of the mo- 

 mental ellipsoid is at the centre of inertia of the rigid 

 body, the directions of the principal axes of the ellipsoid 

 are the same as the principal axes of inertia, and the 

 lengths of the axes of the ellipsoid are inversely propor- 

 tional to the corresponding radii of gyration. When, 

 therefore, the impulsive screw is given, the momental 

 ellipsoid alone must be capable of determining the cor- 

 responding instantaneous screw. 



A family of rigid bodies may be conceived which 

 have a common momental ellipsoid, every rigid body 

 which fulfils nine conditions will belong to this family. 

 If an impulsive wrench applied to a member of this 

 family cause it to twist about a screw 0, then the same 

 impulsive wrench applied to any other member of the 

 same family will cause it likewise to twist about 6. If 

 we added the further condition that the masses of all the 

 members of the family were equal, then it would be found 

 that the twist velocity, and the kinetic energy acquired 

 in consequence of a given impulse, would be the same 

 to whatever member of the family the impulse were 

 applied ( 60, 61). 



146. the Screw Complex of the (n l) '* Order and Se- 

 cond Degree. We shall denote a screw complex of the. 

 n th order and first degree by A, and Oi, . . . O n are the 

 co-ordinates of a screw 6 belonging to A, and referred to 

 n co-reciprocal screws chosen from A . 



Let us first consider the interpretation of the linear 

 equation between the n co-ordinates of 6 : 



