DYNAMICS OF A RIGID BODY. 131 



cipal plane of that cylindroid which contains all the 

 screws of the screw complex which are reciprocal to rj. 



117. Four Screws of the Screw Complex. Take any 

 four screws a, /3, y, S of the screw complex of the third 

 order. Then we shall prove that the cylindroid (a, j3) 

 must have a screw in common with the cylindroid (y, ) 

 For twists of appropriate amplitudes about a, /3, y, S must 

 neutralise, and hence the twists about a, )3 must be coun- 

 teracted by those about y, S ; but this cannot be the 

 case unless there is some screw common to (a, /3) and 



(7, *) 



This theorem provides a convenient test as to whe- 

 ther four screws belong to a screw complex of the third 

 order. 



1 1 8. Equilibrium of Four Forces applied to a Rigid Body. 

 If the body be free, the four forces must be four wrenches 

 on screws of zero pitch which are members of a screw 

 complex of the third order. The forces must therefore 

 be generators of an hyperboloid, and all belonging to 

 the same system ( 106). 



Three of the forces, P, Q, R, being given in position, S 

 must then be a generator of the hyperboloid determined 

 by P, Q y R. This proof of a well-known theorem (due to 

 Mobius) is given to show the facility with which such 

 results flow from the Theory of Screws. 



Suppose, however, that the body have only freedom 

 of the fifth order, we shall find that somewhat more 

 latitude exists with reference to the choice of S. Let X 

 be the screw reciprocal to the screw complex by which 

 the freedom is defined. Then for equilibrium it will 

 only be necessary that S belong to the complex of the 

 fourth order defined by the four screws 



P, Q> R, x. 



A cone of screws can be drawn through eevry point 



K 2 



