258 Prof. Maxwell on Reciprocal Figures 



Another such point can be determined by going round the 

 remaining sides of the polygon ; and these two points, together 

 with the intersections of the lines AE, must all be in one straight 

 line, namely, the intersection of the faces A P and E T. 



Hence the conditions of the possibility of reciprocity in plane 

 figures are the same as those of each figure being the perspective 

 projection of a plane-sided polyhedron. When the number of 

 points is in every part of the figure equal to or less than the 

 number of polygons, this condition is fulfilled of itself. When 

 the number of points exceeds the number of polygons, there will 

 be an impossible case, unless certain conditions are fulfilled so 

 that certain sets of intersections lie in straight lines. 



Application to Statics. 

 The doctrine of reciprocal figures may be treated in a purely 

 geometrical manner, but it may be much more clearly understood 

 by considering it as a method of calculating the forces among a 

 system of points in equilibrium ; for, 



If forces represented in magnitude by the lines of a figure be 

 made to act between the extremities of the corresponding lines 

 of the reciprocal figure, then the points of the reciprocal figure 

 will all be in equilibrium under the action of these forces. 



For the forces which meet in any point are parallel and pro- 

 portional to the sides of a polygon in the other figure. 



If the points between which the forces are to act are known, 

 the problem of determining the relations among the magnitudes 

 of the forces so as to produce equilibrium will be indeterminate, 

 determinate, or impossible, according as the construction of the 

 reciprocal figure is so. 



Reciprocal figures are mechanically reciprocal; that is, either 

 may be taken as representing a system of points, and the other 

 as representing the magnitudes of the forces acting between 

 them. 



In figures like 1, 2 and II., 3 and III., in which the equation 

 e = 2s-2 

 is true, the forces are determinate in their ratios ; so that one 

 being given, the rest may be found. 



When e > 2s — 2, as in figs. 4 and 5, the forces are indetermi- 

 nate, so that more than one must be known to determine the rest, 

 or else certain relations among them must be given, such as those 

 arising from the elasticity of the parts of a frame. 



When e < 2s— 2, the determination of the forces is impossible 

 except under certain conditions. Unless these be fulfilled, as in 

 figs. IV. and V., no forces along the lines of the figure can keep 

 its points in equilibrium, and the figure, considered as a frame, 

 may be said to be loose. 



