DYNAMICS OF A RIGID BODY. 133 



are destroyed by the reactions, and, therefore, the four 

 wrenches on the four screws equilibrate. It is mani- 

 fest that this theorem may be generalised into the fol- 

 lowing: If a body have freedom of the k th order, then 

 properly selected wrenches about any k + i screws (not 

 reciprocal to the screw complex) will hold the body in 

 equilibrium. 



That a rigid body with freedom of the third order 

 may be in equilibrium under the action of gravity, we 

 have the necessary and sufficient condition, which is 

 thus stated : 



The vertical through the centre of inertia must be 

 one of the reciprocal system of generators on the pitch 

 quadric. 



We see that the centre of inertia must, therefore, lie 

 upon a screw of zero pitch which belongs to the screw 

 complex ; whence we have the following theorem : 

 The restraints which are necessary for the equilibrium 

 of a body which has freedom of the third order under 

 the action of gravity, would permit rotation of the body 

 round one definite line through the centre of inertia. 



119. The Ellipsoid of Inertia. The momental ellip- 

 soid, which is of such significance in the theory of the 

 rotation of a rigid body about a fixed point, is presented 

 in the Theory of Screws as a particular case of another 

 ellipsoid called the ellipsoid of inertia, which is of great 

 importance in connexion with the general screw com- 

 plex of the third order. 



If we take three conjugate screws of inertia from the 

 screw complex, as screws of reference, then we have 

 seen (67) that, if ft, ft, ft, be the co-ordinates of a screw 

 $, we have 



where u ly u^ u 3 are the values of u d with reference to the 

 three conjugate screws of inertia. 



