64 THE THEORY OF SCREWS. [72- 



found which are reciprocal to the screw system P. The theory of reciprocal 

 screws will now prove that Q must really be a screw system of order 6 n. 

 In the first place it is manifest that Q must be a screw system of some 

 order, for if a body be capable of twisting about even six independent screws, 

 it must be perfectly free. Here, however, if a body were able to twist about 

 the infinite number of screws embodied in Q, it would still not be free, 

 because it would remain in equilibrium, though acted upon by a wrench 

 about any screw of P. It follows that Q can only denote the collection of 

 screws about which a body can twist which has some definite order of 

 freedom. It is easily seen that that number must be 6 n, for the number 

 of constants disposable in the selection of a screw belonging to a screw 

 system is one less than the order of the system ( 36). But we have seen 

 that the constants disposable in the selection of X are 5 n, and, therefore, 

 Q must be a screw system of order G - n. 



We thus see, that to any screw system P of order n corresponds a reciprocal 

 screw system Q of order 6 n. Every screw of P is reciprocal to all the 

 screws of Q, and vice versa. This theorem provides us with a definite test as 

 to whether any given screw a is a member of the screw system P. Construct 

 6?i screws of the reciprocal system. If then a be reciprocal to these 6 n 

 screws, a must in general belong to P. We thus have 6 n conditions to 

 be satisfied by any screw when a member of a screw system of order n. 



73. Equilibrium. 



If the screw system P expresses the freedom of a rigid body, then the 

 body will remain in equilibrium though acted upon by a wrench on any 

 screw of the reciprocal screw system Q. This is, perhaps, the most general 

 theorem which can be enunciated with respect to the equilibrium of a rigid 

 body. This theorem is thus proved : Suppose a wrench to act on a screw 77 

 belonging to Q. If the body does not continue at rest, let it commence to 

 twist about a. We would thus have a wrench about rj disturbing a body 

 which twists about a, but this is impossible, because a and ?; are reciprocal. 



In the same manner it may be shown that a body which is free to twist 

 about all the screws of Q will not be disturbed by a wrench about any screw 

 of P. Thus, of two reciprocal screw systems, each expresses the locus of a 

 wrench which is unable to disturb a body free to twist about any screw of 

 the other. 



74. Reaction of Constraints. 



It also follows that the reactions of the constraints by which the move 

 ments of a body are confined to twists about the screws of a system P can 

 only be wrenches on the reciprocal screw system Q, for the reactions of the 



