24 GEOMETRY OF VECTORS. [CHAP. II. 



side of the first line which makes the moment have the right sign. 

 The figures a and j3 show this for the two cases. 



Fig. 20. 



When P and Q have like senses the system is equivalent to a 

 vector of magnitude P + Q localised in a line parallel to the lines 

 of P and Q, between these lines, and at a distance d' from the 

 line of P such that (P + Q) d' = Qd. The sense of this vector is 

 that of P and Q. 



When P and Q have opposite senses, and Q > P, the system is 

 equivalent to a vector of magnitude Q P localised in a line 

 parallel to the line of P or Q, on the side of the line of Q remote 

 from the line of P, and at a distance d' from the line of P such 

 that (Q - P) d' = Qd. The sense of this vector is that of Q. 



It is clear that in both cases the parallel vectors have a single 

 resultant, localised in a parallel line, and of such magnitude and 

 sense that its moment about any point in the plane of the two 

 parallel vectors is equal to the sum of the moments of the two 

 parallel vectors about the same point. 



The particular case of this theorem required in our proof of 

 Theorem II. is that the resultant of two vectors of equal magnitude 

 and like sense, localised in parallel lines, is a vector of twice the 

 magnitude and of the same sense, localised in a line midway 

 between the two parallels. 



23. Theorem II. Two couples in parallel planes are equi- 

 valent if they have the same moment. 



We shall prove that two couples in parallel planes of equal 

 moments in opposite senses are equivalent to zero. 



Let the vectors of one couple be of magnitude P, and be 



