40 EQUILIBRIUM OF A RIGID BODY. 



plex. For, by the definition of the screw complex, it ap- 

 pears that twists of appropriate amplitudes about A ,, &c., 

 A n , would compound into a twist about A. It fol- 

 lows ( 43) that wrenches on AU &c., A n , of appropriate 

 intensities (32) compound into a wrench on A . Suppose 

 these wrenches on A\ y &c., A n , were applied to a body 

 only free to twist about X, then since X is reciprocal to 

 A i, &c., A n , the equilibrium of the body would be un- 

 disturbed. The resultant wrench on A must therefore 

 be incapable of moving the body, therefore^ and X must 

 be reciprocal. 



46. The Reciprocal Screw Complex. All the screws 

 which are reciprocal to a screw complex P of order k 

 constitute a screw complex Q of order 6 - k. This im- 

 portant theorem is thus proved : 



Since only one condition is necessary for a pair of 

 screws to be reciprocal, it follows, from the last section, 

 that if a screw X be reciprocal to P it will fulfil k con- 

 ditions. The screw X has, therefore, 5 - k elements still 

 disposable, and consequently (k < 5) an infinite number 

 of screws Q can be found which are reciprocal to the 

 screw complex P. The theory of reciprocal screws will 

 now prove that Q must really be a screw complex of 

 order 6 - k. In the first place it is manifest that Q must 

 be a screw complex of some order, for, in general, if a 

 body be capable of twisting about even six screws, it 

 must be perfectly free. Here, however, if a body were 

 able to twist about the infinite number of screws em- 

 bodied in Q y it would still not be free, because it would 

 remain in equilibrium, though acted upon by a wrench 

 about any screw of P. If 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 - k, for the number of 

 constants disposable in the selection of a screw belong- 



