RECIPROCAL FIGURES, FRAMES, AND DIAGRAMS OF FORCES. 3 



corresponding lines of the diagram of forces should be indicated by the same 

 symbol, accented if necessary. 



We have supposed the corresponding lines to be parallel, and it is necessary 

 that they should be parallel when the frame is not in one plane ; but if all the 

 pieces of the frame are parallel to one plane, we may turn one of the diagrams 

 round a right angle, and then every line will be perpendicular to the corres- 

 ponding line. 



If any number of lines meet at the same point in the frame, the correspond- 

 ing lines in the diagram of forces form a closed polygon. 



It is possible, in certain cases, to draw the diagram of forces so that if any 

 number of lines meet in a point in the diagram of forces, the corresponding lines 

 in the frame form a closed polygon. 



In such cases, the two diagrams are said to be reciprocal in the sense in 

 which we use it in this paper. If either diagram be taken as representing 

 the frame, the lines of the other diagram will represent a system of forces 

 which, if applied along the corresponding pieces of the frame, will keep it in 

 equilibrium. 



The properties of the " triangle " and " polygon " of forces have been long 

 known, and a " 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 proposition that forces acting on a point perpendicular and proportional to 

 the areas of the faces of a polyhedron are in equilibrium, has, I believe, been 

 enunciated independently at various times, but the application of this principle 

 to the construction of a diagram of forces in three dimensions was first made 

 by Professor Eankine in the "Philosophical Magazine," Feb. 1864. In the 

 "Philosophical Magazine" for April 1864, I stated some of the properties of 

 reciprocal figures, and the conditions of their existence, and showed that any 

 plane rectilinear figure which is a perspective representation of a closed poly- 

 hedron with plane faces has a reciprocal figure. In Sept. 1867, I communi- 

 cated to the British Association a method of drawing the reciprocal figure, 

 founded on the theory of reciprocal polars. 



I have since found that the construction of diagrams of forces in which each 

 force is represented by one line, had been independently discovered by Mr W. 

 P. Taylor, and had been used by him as a practical method of determining 

 the forces acting in frames for several years before I had taught it in King's 

 College, or even studied it myself. I understand that he is preparing a state- 

 ment of the application of the method to various kinds of structures in detail, 

 so that it can be made use of by any one who is able to draw one fine parallel 

 to another. 



