13-] 



THEORY OF COUPLES. 



77 



Fig. 31. 



two equal and parallel forces F at A and B 1 form a resultant 

 2F at the middle point O of the 

 diagonal AB' of the parallelogram 

 ABB' A'. Similarly, the two forces 

 FatB and A' are together equiva- | 

 lent to a resultant 2 Fat the same 

 point O. These two resultants, be- 

 ing equal and opposite and acting in 

 the same line, are together equiva- 

 lent to o. Hence the whole system 

 reduces to the force F* at A' and the force F at B\ which 

 form, therefore, a couple equivalent to the original couple at 

 AB. 



130. The effect of a couple is not changed by rotation in its 

 plane. 



Let AB (Fig. 32) be the arm of the couple in the original 

 position, C its middle point, and let the couple be turned about 

 C into the position AB 1 . Applying again at A', B 1 equal and 

 opposite forces each equal to F, the forces .Fat A' and Fat A 

 will form a resultant acting along CD, while Fat B 1 and F'at 

 B give an equal and opposite resultant along CE. These two 



resultants destroy each other 

 and leave nothing but the 

 couple formed by F at A' and 

 F at B', which is therefore 

 equivalent to the original 

 couple. 



Any other displacement of 

 the couple in its plane, or to a 

 parallel plane, can be effected 

 by a translation combined with 

 a rotation about the middle 

 point of its arm in its plane. 

 The effect of a couple is therefore not changed by any displace- 

 ment in its plane or to a parallel plane. 



