76] THE EQUILIBRIUM OF A RIGID BODY. 65 



constraints are only manifested by the success with which they resist the 

 efforts of certain wrenches to disturb the equilibrium of the body. 



75. Parameters of a Screw System. 



We next consider the number of parameters required to specify a screw 

 system of the ?ith order often called for brevity an ?i-system. Since the 

 system is defined when n screws are given, and since five data are required 

 for each screw, it might be thought that on parameters would be necessary. 

 It must be observed, however, that the given 5n data suffice not only for the 

 purpose of defining the screw system but also for pointing out n special 

 screws upon the screw system, and as the pointing out of each screw on the 

 system requires n - 1 quantities ( 69), it follows that the number of 

 parameters actually required to define the system is only 



5n n (n 1) = n (6 n). 



This result has a very significant meaning in connexion with the theory 

 of reciprocal screw systems 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 system is identical with the number necessary to specify its reciprocal 

 screw system. This remark is chiefly of importance in connexion with the 

 systems of the fourth and fifth orders, which are respectively the reciprocal 

 systems of a cylindroid and a single screw. We are now assured that a 

 collection of all the screws which are reciprocal to an arbitrary cylindroid can 

 be nothing less than a screw system of the fourth order in its most general 

 type, and also, that all the screws in space which are reciprocal to a single 

 screw must form the most general type of a screw system of the fifth order. 



76. Applications of Co-ordinates. 



If the co-ordinates of a screw satisfy n linear equations, the screw must 

 belong to a screw system of the order 6 n. Let 77 be the screw, and let one 

 of the equations be 



whence 77 must be reciprocal to the screw whose co-ordinates are pro 

 portional to 



^, . ~ 6 ,(37). 



Pi P* 



It follows that 77 must be reciprocal to n screws, and therefore belong to a 

 screw system of order 6 n. 



Let a, /3, 7, B be for example four screws about which a body receives 

 twists of amplitudes a , # , 7 , & . It is required to determine the screw p and 

 B. 5 



