_Reciprocal Diagrams._--When to every point of concourse of the lines in the diagram of stress corresponds a closed polygon in the skeleton of the frame, the two diagrams are said to be reciprocal.
The first extensions of the method of diagrams of forces to other cases than that of the funicular polygon were given by Rankine in his _Applied Mechanics_ (1857). The method was independently applied to a large number of cases by W. P. Taylor, a practical draughtsman in the office of J. B. Cochrane, and by Professor Clerk Maxwell in his lectures in King's College, London. In the _Phil. Mag._ for 1864 the latter pointed out the reciprocal properties of the two diagrams, and in a paper on "Reciprocal Figures, Frames and Diagrams of Forces," _Trans. R.S. Edin._ vol. xxvi., 1870, he showed the relation of the method to Airy's function of stress and to other mathematical methods. Professor Fleeming Jenkin has given a number of applications of the method to practice (_Trans. R.S. Edin._ vol. xxv.).
L. Cremona (_Le Figure reciproche nella statica grafica_, 1872) deduced the construction of reciprocal figures from the theory of the two components of a wrench as developed by Möbius. Karl Culmann, in his _Graphische Statik_ (1st ed. 1864-1866, 2nd ed. 1875), made great use of diagrams of forces, some of which, however, are not reciprocal. Maurice Levy in his _Statique graphique_ (1874) has treated the whole subject in an elementary but copious manner, and R. H. Bow, in his _The Economics of Construction in Relation to Framed Structures_ (1873), materially simplified the process of drawing a diagram of stress reciprocal to a given frame acted on by a system of equilibrating external forces.
[Illustration: FIG. 1.--Diagram of Configuration.]
Instead of lettering the joints of the frame, as is usually done, or the links of the frame, as was the custom of Clerk Maxwell, Bow places a letter in each of the polygonal areas enclosed by the links of the frame, and also in each of the divisions of surrounding space as separated by the lines of action of the external forces. When one link of the frame crosses another, the point of apparent intersection of the links is treated as if it were a real joint, and the stresses of each of the intersecting links are represented twice in the diagram of stress, as the opposite sides of the parallelogram which corresponds to the point of intersection.