ON RECIPROCAL FIGURES AND DIAGRAMS OF FORCES. 



517 



If any number of the points of the figure are so connected among them- 

 selves as to form an equal number of closed polygons, the conditions of 

 constructing the reciprocal figure must be found by considering these points 

 separately, and then examining their connexion with the rest. 



Let us now consider a few cases of reciprocal figures in detail. The 

 simplest case is that of the figure formed by the six lines connecting four 

 points in a plane. If we now draw the six lines con- 

 necting the centres of the four circles which pass through 

 three out of the four points, we shall have a reciprocal 

 figure, the corresponding lines hi the two figures being 

 at right angles. 



The reciprocal figure formed in this way is definite 

 in size and position ; but any figure similar to it and 

 placed in any position is still reciprocal to the original 

 figure. If the reciprocal figures are lettered as in fig. 1, 

 we shall have the relation 



Fig. 1. 



AP 

 ap 



BQ CR 



cr 



In figures 2 and II. we have a pair of reciprocal figures in which the 

 lines are more numerous, but the construction very easy. There are seven 

 points in each figure corresponding to seven polygons in the other. 



Fig. 2. 



Fig. II. 



The four points of triple concourse of lines ABC, BDE, HIL, LJK 

 correspond to four triangles, abc, bde, hil, Ijk. 



The three points of quadruple concourse ADFH, CEGK, IFGJ correspond 

 to three quadrilaterals, adfh, cegk, ifgj. 



The five triangles ADB, EBC, GJK, IJL, HIF correspond to five points 

 of triple concourse, adb, ebc, gjk, ijl, hif. 



