132 DYNAMICS OF A RIGID BODY. 



in space belonging to this complex, and on that cone 

 one screw of zero pitch can always be found. Hence one 

 line can be drawn through every point in space along 

 which S might act. 



If the body have freedom of the fourth order, the lati- 

 tude in the choice of S is still greater. Let X ly X z be 

 two screws reciprocal to the complex, then S is only 

 restrained by the condition that it belong to the screw 

 complex of the fifth order defined by the screws 



P, Q, X, X 19 X,. 



Any line in space when it receives the proper pitch 

 is a screw of this complex. 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 complex ( 80). 



Finally, if the body has only freedom of the third 

 order, the four equilibrating forces P, Q, R, S may be si- 

 tuated anywhere. 



The positions of the forces being given, their magni- 

 tudes are determined ; for draw three screws X ly X z , X$ 

 reciprocal to the complex, and find ( 30) the intensities 

 of the seven equilibrating wrenches on 



The last three are neutralised by the reactions of the 

 constraints, and the four former must therefore equili- 

 brate. 



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 



