and Diagrams of Forces. 251 



perpendicular, or at any constant angle. Lines meeting in a 

 point in one figure form a closed polygon in the other. 



In figures in space, the lines in one figure are perpendicular 

 to planes in the other, and the planes corresponding to lines 

 which meet in a point form a closed polyhedron. 



The conditions of reciprocity may be considered from a purely 

 geometrical point of view ; but their chief importance arises from 

 the fact that either of the figures being considered as a system of 

 points acted on by forces along the lines of connexion, the other 

 figure is a diagram of forces, in which these forces are represented 

 in plane figures by lines, and in solid figures by the areas of 

 planes. 



The properties of the " triangle " and " polygon " of forces 

 have been long known, and the " diagram " of forces has been 

 used in the case of the funicular polygon ; but I am not aware of 

 any more general statement of the method of drawing diagrams 

 of forces before Professor Rankine applied it to frames, roofs, 

 &c. in his ' Applied Mechanics/ p. 137, &c. The "polyhedron 

 of forces," or the equilibrium of forces perpendicular and pro- 

 portional to the areas of the faces of a polyhedron, has, I believe, 

 been enunciated independently at various times ; but the appli- 

 cation to a "frame" is given by Professor Rankine in the Phi- 

 losophical 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 con- 

 sist of an equal number of lines, so that corresponding lines in 

 the two figures are parallel, and corresponding lines which con- 

 verge 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 j 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 poly- 

 gons. 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. 



