ON RECIPROCAL FIGURES AND DIAGRAMS OF FORCES. 515 



of drawing diagrams of forces before Professor Eankine applied it to frames, 

 roofs, &c. in his Applied Mechanics, p. 137, &c. The "polyhedron of forces," 

 or the equilibrium of forces perpendicular and proportional to the areas of the 

 faces of a polyhedron, has, I believe, been enunciated independently at various 

 times; but the application to a "frame" is given by Professor Rankine in the 

 Philosophical Magazine, February, 1864. 



I propose to treat the question geometrically, as reciprocal figures are 

 subject to certain conditions besides those belonging to diagrams of forces. 



On Reciprocal Plane Figures. 



Definition. Two plane figures are reciprocal when they consist of an equal 



number of lines, so that corresponding lines in the two figures are parallel, 



and corresponding lines which converge to a point in one figure form a closed 

 polygon in the other. 



Note, If corresponding lines in the two figures, instead of being parallel 

 are at right angles or any other angle, they may be made parallel by turning 

 one of the figures round in its own plane. 



Since every polygon in one figure has three or more sides, every point in 

 the other figure must have three or more lines converging to it ; and since 

 every line in the one figure has two and only two extremities to which lines 

 converge, every line in the other figure must belong to two, and only two 

 closed polygons. The simplest plane figure fulfilling these conditions is that 

 formed by the six lines which join four points in pairs. The reciprocal figure 

 consists of six lines parallel respectively to these, the points in the one figure 

 corresponding to triangles in the other. 







General Relation between the Numbers of Points, Lines, and Polygons in 



Reciprocal Figures. 



The effect of drawing a line, one of whose extremities is a point connected 

 with the system of lines already drawn, is either to introduce one new point 

 into the system, or to complete one new polygon, or to divide a polygon into 

 two parts, according as it is drawn to an isolated point, or a point already 

 connected with the system. Hence the sum of points and polygons in the 



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