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22 GEOMETRY OF VECTORS. [CHAP. II. 



to the plane of the couple, and of length containing a number of 

 units of length equal to the number which expresses the moment 

 of the couple, is called the axis of the couple. The sense in which 

 the line is drawn is that of a parallel line meeting the line of 

 one of the vectors, and such that the moment of the other about 

 it is positive. 



The series of three theorems we are now going to prove lead 

 to the result that a couple is equivalent to an unlocalised vector 

 represented by its axis. 



21. Theorem I. Two couples in the same plane are equivalent 

 if they have the same moment. 



We shall prove that two couples in the same plane, of equal 



moments, in opposite senses, 

 are equivalent to zero. 



The lines in which the vec- 

 tors are localised, being two- 

 pairs of parallel lines, form a 

 parallelogram. Let this be 

 ABCD. 



Let the vectors of one 

 couple be of magnitude P, 

 Flg * 18< and be localised in the lines 



AB, CD\ and let the vectors of the other couple be of magnitude 

 Q, and be localised in the lines AD, CB. 



Let the unit of length be so chosen that AB represents P in 

 magnitude. 



Then the area of the parallelogram is of magnitude equal to 

 the moment of the couple. 



Hence AD represents Q in magnitude. 



Now the vectors P and Q localised in the lines AB, AD, and 

 proportional to those lines, are equivalent to a vector localised 

 in the line AC, and proportional to that line. The sense of this 

 vector is AC. 



Also the vectors P and Q localised in the lines CD, CB, and 

 proportional to those lines, are equivalent to a vector localised in 

 the line CA, and proportional to that line. The sense of this 

 vector is CA. 



