DYNAMICS OF A RIGID BODY. 87 



If two wrenches act upon the body, then the condi- 

 tion of equilibrium is, that, when the two wrenches are 

 compounded by the aid of the cylindroid, the single 

 wrench which replaces them shall lie upon that one 

 screw of the cylindroid, which is reciprocal to a ( 28). 



We can express with great facility, by the aid of 

 screw co-ordinates, the condition that wrenches of in- 

 tensities 0", 0", on two screws 0, 0, shall equilibrate, 

 when applied to a body only free to twist about a. 



Adopting any six co-reciprocals as screws of refer- 

 ence, and resolving each of the wrenches on 9 and <p 

 into its six components on the six screws of reference, 

 we shall have for the intensity of the component of the 

 resultant wrench on wn 



Hence the co-ordinates of the resultant wrench are pro- 

 portional to 



V 0i + 01, ... 06 + 06 



For equilibrium this screw must be reciprocal to a, 

 whence we have 



0"0,) + &C. + / 6 a 6 (0"0 6 + 0"0 6 ) = O, 



or, 



0X, + 0X* = o. 



This equation merely expresses that the sum of the 

 works done in a small twist about a against the wrenches 

 on and is zero. 



We also perceive that a given wrench may be always 

 replaced by a wrench of appropriate intensity on any 

 other screw, in so far as the effect on a body only free to 

 twist about a is concerned. 



It may not be out of place to notice the analogy 

 which the equation just written bears to the simple prob- 



