and Diagrams of Forces. 



253 



If s </, the construction of the reciprocal diagram will be 

 impossible unless (s— •/) conditions be fulfilled in the original 

 diagram. 



If any number of the points of the figure are so connected 

 among themselves as to form an equal number of closed poly- 

 gons, the conditions of constructing the reciprocal figure must 

 be found by considering these points separately, and then ex- 

 amining 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 

 connecting the centres of the four circles which pass through 

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

 the corresponding lines in the two figures being at right angles. 



The reciprocal figure formed in p- ^ 



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 



AP = BQ = CR 



ap bq cr 



9/ 



X* c 





k» V 









p 







p / 







c\ 



/b 





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. Fie. II. 



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

 H I L, L J K correspond to four triangles, a b c, b de, hi I, Ij k. 



The three points of quadruple concourse ADFH, CEGK, 

 IFGJ correspond to three quadrilaterals, adfh, cegk, ifgj. 



