186] FREEDOM OF THE THIRD ORDER. 187 



Any line in space when it receives the proper pitch is a screw of this 

 system. Through any point in space a plane can be drawn such that every 

 line in the plane passing through the point with zero pitch is a screw of the 

 system ( 110). 



Finally, if the body has only freedom of the third order, the four equi 

 librating forces P, Q, R, S may be situated anywhere. 



The positions of the forces being given, their magnitudes are determined ; 

 for draw three screws X lt X. 2&amp;gt; X 3 reciprocal to the system, and find ( 28) the 

 intensities of the seven equilibrating wrenches on 



4j Q&amp;gt; -R&amp;gt; &amp;gt;&amp;gt; Xj, X 2 , X 3 . 



The last three are neutralised by the reactions of the constraints, and 

 the four former must therefore equilibrate. 



Given any four screws in space, it is possible for four wrenches of proper 

 intensities on these screws to hold a body having freedom of the third order 

 in equilibrium. For, take the four given screws, and three reciprocal screws. 

 Wrenches of proper intensities on these seven screws will equilibrate ; but 

 those on the reciprocal screws are destroyed by the reactions, and, therefore, 

 the four wrenches on the four screws equilibrate. It is manifest that this 

 theorem may be generalised into the following : If a body have freedom of 

 the kth order, then properly selected wrenches about any k+l screws (not 

 reciprocal to the screw system) 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 system ; 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. 



186. The Ellipsoid of Inertia. 



The momental ellipsoid, 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 

 system of the third order. 



