ON RECIPROCAL FIGURES AND DIAGRAMS OF FORCES. 



523 



finitely great forces in some of the pieces, or may throw the frame into a loose 

 condition at once. 



The conditions, however, of the possibility of determining the ratios of the 

 forces in a frame are not coextensive with those of finding a figure perfectly 

 reciprocal to the frame. The condition of determinate forces is 



e = 2s-2; 



the condition of reciprocal figures is that every line belongs to two polygons 

 only, and 



In fig. 7 we have six points connected by ten lines in such a way that 

 the forces are all determinate ; but since the line L is a ,side of three triangles, 

 we cannot draw a reciprocal figure, for we should have to draw a straight line 

 I with three ends. 



If we attempt to draw the reciprocal figure as in fig. VIL, we shall find 

 that, in order to represent the reciprocals of all the lines of fig. 7 and fix 

 their relations, we must repeat two of them, as h and e by h' and e, so as 

 to form a parallelogram. Fig. VII. is then a complete representation of the rela- 

 tions of the force which would produce equilibrium in fig. 7 ; but it is redundant 

 by the repetition of h and e, and the two figures are not reciprocal. 



Fig. VII. 



Fig. 7. 



On Reciprocal Figures in three dimensions. 



Definition. Figures in three dimensions are reciprocal when they can be so 

 placed that every line in the one figure is perpendicular to a plane face of the 

 other, and every point of concourse of lines in the one figure is represented by 

 a closed polyhedron with plane faces. 



662 



