EQUILIBRIUM OF A RIGID BODY. 41 



ing to a screw complex is one less than the order of the 

 complex (38). But. we have seen that the constants dis- 

 posable in the selection of X are 5 - k, and, therefore, Q 

 must be a screw complex of order 6 - k. 



We thus see, that to any screw complex P of order k be- 

 longs a reciprocal screw complex Q of order 6 - k. 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 

 complex P. Construct any 6 - k screws of the reciprocal 

 system. If then a be reciprocal to these 6 - k screws, a 

 must belong to P. We thus have 6 - k conditions to be 

 satisfied by any screw when a member of a screw com- 

 plex of order k. 



47. Equilibrium. If the screw complex 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 complex 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 rj belonging to Q. If the body does 

 not continue at rest, let it commence to twist about a. 

 We thus have a wrench about TJ disturbing a body which 

 twists about a, but this is impossible, because a and j 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 complexes, each expresses the 

 locus of a wrench which is unable to disturb a body free 

 to twist about any screw of the other. 



48. Reaction of Constraints. It also follows that the 

 reactions of the constraints by which the movements 

 of a body are confined to twists about the screws of 



