73 STATICS. [131. 



131. The effect of a couple is not changed if its force F and its 

 arm p be cJianged simultaneously in any way, provided their 

 product Fp remain the same. 



Let AB=p be the original arm (Fig. 33), F the original force 

 of the couple ; and let A''=J>' be the new arm. The introduc- 

 tion of two equal and opposite forces F' at A', and also at B\ 

 will not change the given system F, F. Now, selecting for 



F' a magnitude such that F'p' = Fp, 

 the force F at A and the force F 1 

 F and A' combine (Arts. 104-106) to 

 A' form a parallel resultant through C, 

 the middle point of the arm, since 



B 





--B i- 



-F 



Fig. 33. 



_ F ' 



for this point F- J 



Similarly, -F at B and /?' at .#' 



give a resultant of the same magni- 



tude, in the same line through C, but of opposite sense. 



These two resultants thus destroying each other, there remains 



only the couple formed by F 1 at A' and F' at B\ for which 



Fp = F'p'. 



132. It results from the last three articles that the only essen- 

 tial characteristics of a couple are (a) the numerical value of the 

 moment ; (b) the sense, or direction of rotation ; and (c) what 

 has been called the "aspect" of its plane, i.e. the direction of 

 any normal to this plane. 



It is to be noticed that the plane of the two forces forming 

 the couple is not an essential characteristic of the couple ; just 

 as the point of application of a force is not an essential charac- 

 teristic of the force (see Art. 84). 



Now the three characteristics enumerated above can all be 

 indicated by a vector which can therefore serve as the geomet- 

 rical representative of the couple. Thus, the couple formed 

 by the forces F, F (Fig. 34), whose perpendicular distance 

 is /, is represented by the vector AB = Fp laid off on any 

 normal to the plane of the couple. The sense is indicated by 



