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'. 



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 r , C', 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. 



