22, 23] 



COUPLES IN PARALLEL PLANES. 



25 



localised in the lines AB, CD\ and let the vectors of the other 

 couple be of magnitude Q, and be localised in the lines A D , C B . 



c 



Fig. 21. 



Through A D and B C draw a pair of parallel planes meeting 

 the lines of the couple P in the points A, D, B, C. 



Through AB and CD draw a pair of parallel planes meeting 

 the lines of the couple Q in the points A , B, (7, D . 



These two pairs of planes with the planes of the two couples 

 form a parallelepiped. 



Replace the couple Q in its plane by an equivalent couple 

 consisting of vectors localised in the lines B A and D C . These 

 vectors are both of magnitude P, and have the senses indicated 

 by the order of the letters. 



Now parallel vectors P localised in lines AB, D C , and having 

 the senses indicated, are equivalent to a vector of magnitude 2P 

 localised in the line MM joining the middle points of AD and 

 BC . The sense of this vector is MM . 



Also parallel vectors P localised in lines CD, B A are equi 

 valent to a vector of magnitude 2P localised in the same line 

 MM . The sense of this vector is M M. 



It follows that the set of four vectors P, P, and Q, Q are equi 

 valent to zero. 



This theorem shows that a couple may be replaced by any 

 couple of the same moment in any parallel plane. 





