26 GEOMETRY OF VECTORS. [CHAP. II. 



24. Theorem III. Couples obey the vector law of composition. 



C' 



A p B 



Fig. 22. 



Let the planes of the couples meet in the line AB. 



Replace the couple in one plane by any couple having one of 

 its vectors localised in AB in the sense AB. 



Let the two vectors be of magnitude P, and let the other be 

 localised in the line CD. 



Replace the couple in the other plane by a couple having one 

 of its vectors localised in BA in the sense BA. 



We can take these vectors also to be of magnitude P, and then 

 the other will be localised in a certain line FE in the plane of 

 the second couple. 



Let AB represent P in magnitude, and through the points A, 

 B draw planes at right angles to AB cutting the lines CD and 

 EF in the points named C } D, E, F. 



Then the two couples are seen to be equivalent to a single 

 couple, whose vectors are of magnitude P, and are localised in the 

 lines CD, FE. 



The figures ABCD, ABEF, CDFE are rectangles, and their 

 areas are proportional to the moments of the couples. These 

 areas are in the ratios of the lengths of BC, BE, CE. 



Hence if we turn the triangle BCE through a right angle in 

 its plane its sides will be parallel and proportional to the axes of 

 the couples. Let B'C'E' be the new triangle; then it is clear 

 that if E'B' represents the axis of the second couple in sense, the 

 sense of the first is B'C', and the sense of the resultant is E'C'. 



Thus the axis of a couple which has the magnitude, direction, 

 and sense of a line E'C' is the axis of the resultant of two com- 

 ponent couples, the axes of the components having the magnitudes, 



