[From the London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science. 



Vol. xxvii. Fourth Series.] 



XXIV. On Reciprocal Figures and Diagrams of Forces. 



RECIPROCAL figures are such that the properties of the first relative to the 

 second are the same as those of the second relative to the first. Thus inverse 

 figurea and polar reciprocals are instances of two different kinds of reciprocity. 



The kind of reciprocity which we have here to do with has reference to 

 figures consisting of straight lines joining a system of points, and forming 

 closed rectilinear figures ; and it consists in the directions of all lines in the 

 one figure having a constant relation to those of the lines in the other figure 

 which correspond to them. 



In plane figures, corresponding lines may be either parallel, 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 



