522 ON RECIPROCAL FIGURES AND DIAGRAMS OF FORCES. 



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 proportional 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>2 2, as in figs. 4 and 5, the forces are indeterminate, 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. 



When the conditions are fulfilled, the pieces of the frame can support forces, 

 but in such a way that a small disfigurement of the frame may produce in- 



