ON RECIPROCAL FIGURES AND DIAGRAMS OF FORCES. 



519 



In figures 4 and IV. the condition that the number of polygons is equal 

 to the number of points is not fulfilled. In fig. 4 there are five points and 



Fig. IV. 



Fig. 4. 



six triangles ; in fig, IV. there afe six points, two triangles, and three quadri- 

 laterals. Hence if fig. 4 is given, fig. IV. is indeterminate to the extent of one 

 variable, besides the elements of scale and position. In fact when we have drawn 

 ABC and indicated the directions of P, Q, R, we may fix on any point of P 

 as one of the angles of X YZ and complete the triangle XYZ. The size of 

 XYZ is therefore indeterminate. Conversely, if fig. IV. is given, fig. 4 cannot 

 be constructed unless one condition be fulfilled. That condition is that P, Q, 

 and R meet in a point. When' this is fulfilled, it follows by geometry that 

 the points of concourse of A and X, B and Y, and C and Z lie in one straight 

 line W, which is parallel to w in fig. 4. The condition may also be expressed 

 by saying that fig. IV. must be a perspective projection of a polyhedron whose 

 quadrilateral faces are planes. The planes of these faces intersect at the concourse 

 of P, Q, R, and those of the triangular faces intersect in the line W. 



Figs. 5 and V. represent another case of the same kind. In fig. 5 we 

 have six points and eight triangles ; fig. V. is therefore capable of two degrees 

 of variability, and is subject to two conditions. 



