42 EQUILIBRIUM OF A RIGID BODY. 



a complex P can only be wrenches on the reciprocal 

 screw complex Q, for the reactions of the constraints ar& 

 only manifested by the success with which they resist 

 the efforts of certain wrenches to disturb the equilibrium 

 of the body. 



49. Parameters of a Screw Complex. We next con- 

 sider the question as to how many parameters are 

 required in order to specify completely a screw com- 

 plex of the n th order. Since the complex is defined 

 when n screws are given, and since five data are required 

 for each screw, it might be thought that $n parameters 

 would be necessary. It must be observed, however, that 

 the given $n data suffice not only for the purpose of de- 

 fining the screw complex but also for pointing out n 

 special screws upon the screw complex, and as the point- 

 ing out of each screw on the complex requires n i 

 quantities ( 38), it follows that the number of parameters, 

 actually required to define the complex is only 



$n- n(n- i) = n (6 - n). 



This result has a very significant meaning in con- 

 nexion with the theory of reciprocal screw complexes P 

 and Q. Assuming that the order of P is n, the order of 

 Q is 6 - n ; but the expression n (6 -n ) is unaltered by 

 changing n into 6 - n. It follows that the number of 

 parameters necessary to specify a screw complex is 

 identical with the number necessary to specify its reci- 

 procal screw complex. This remark is chiefly of impor- 

 tance in connexion with the complexes of the fourth and 

 fifth orders, which are respectively the reciprocal com- 

 plexes of a cylindroid and a single screw. We are now 

 assured that a collection of all the screws which are re- 

 ciprocal to an arbitrary cylindroid can be nothing less 

 than a screw complex of the fourth order in its most 

 general type, and also, that all the screws in space which 



