[From the Transactions of the Royal Society of Edinburgh, Vol. xxvi.] 



XXXIX. On Reciprocal Figures, Frames, and Diagrams of Forces. 

 (Received 17th Dec. 1869; read 7th Feb. 1870.) 



Two figures are reciprocal when the properties of the first relative to the 

 second are the same as those of the second relative to the first. Several kinds 

 of reciprocity are known to mathematicians, and the theories of Inverse Figures 

 and of Polar Reciprocals have been developed at great length, and have led to 

 remarkable results. I propose to investigate a different kind of geometrical 

 reciprocity, which is also capable of considerable development, and can be ap- 

 plied to the solution of mechanical problems. 



A Frame may be defined geometrically as a system of straight lines con- 

 necting a number of points. In actual structures these lines are material pieces, 

 beams, rods, or wires, and may be straight or curved ; but the force by which 

 each piece resists any alteration of the distance between the points which it joins 

 acts in the straight line joining those points. Hence, in studying the equilibrium 

 of a frame, we may consider its different points as mutually acting on each 

 other with forces whose directions are those of the lines joining each pair of 

 points. When the forces acting between the two points tend to draw them 

 together, or to prevent them from separating, the action along the joining line 

 is called a Tension. When the forces tend to separate the points, or to keep 

 them apart, the action along the joining line is called a Pressure. 



If we divide the piece joining the points by any imaginary section, the 

 resultant of the whole internal force acting between the parts thus divided will 

 be mechanically equivalent to the tension or pressure of the piece. Hence, in 

 order to exhibit the mechanical action of the frame in the most elementary 

 manner, we may draw it as a skeleton, in which the different points are joined 



VOL. II. 21 



