5] TWISTS AND WRENCHES. 9 



movement was simply a translation. But even in this case it would be 

 impossible for the points Q and R to be distinct from Q. 2 and R 2 , because, 



Fig. l. 



when a body is translated so that all its points move in parallel lines, it is 

 impossible, if the body be rigid, for the distances traversed by each point not 

 to be all equal. We have thus demonstrated that if a body is free to 

 move from a position A 1 to an adjacent position A n by an infinitely small 

 but continuous movement, it is also free to move through the series of 

 positions A. 2&amp;gt; A 3 , &c., by which it would be conveyed from A^ to A n by a twist. 



We may also state the matter in a somewhat different manner, as 

 follows : It would be impossible to devise a system of constraints which 

 would permit a body to be moved continuously from A 1 to A n , and would at the 

 same time prohibit the body from twisting about the screw which directly 

 conducts from A l to A n . Of course this would not be true except in the case 

 where the motion is infinitely small. The connexion of this result with the 

 present investigation is now obvious. When A is the standard position of the 

 body, and B an adjacent position into which it can be moved, then the body 

 is free to twist about the screw defined by A and B. 



5. The canonical form of a small displacement. 



In the Theory of Screws we are only concerned with the small displace 

 ments of a system, and hence we can lay down the following fundamental 

 statement. 



The canonical form to which the displacement of a rigid body can be 

 reduced is a twist about a screw. 



If a body receive several twists in succession, then the position finally 

 attained could have been reached in a single twist, which is called the 

 resultant tivist. 



