164 RECIPROCAL FIGURES, FRAMES, 



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

 of reciprocal figures, and the conditions of their existence, and shewed that 

 any plane rectilinear figure which is a perspective representation of a closed 

 polyhedron 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 deter- 

 mining 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 statement 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 

 line parallel to another. 



Professor Fleeming Jenkin, in a paper recently published by the Society, 

 has fully explained the application of the method to the most important cases 

 occurring in practice. 



In the present paper I propose, first, to consider plane diagrams of frames 

 and of forces in an elementary way, as a practical method of solving questions 

 about the stresses in actual frameworks, without the use of long calculations. 



I shall then discuss the subject in a theoretical point of view, and give a 

 method of defining reciprocal diagrams analytically, which is applicable to 

 figures either of two or of three dimensions. 



Lastly, I shall extend the method to the investigation of the state of 

 stress in a continuous body, and shall point out the nature of the function of 

 stress first discovered by the Astronomer Royal for stresses in two dimensions, 

 extending the use of such functions to stresses in three dimensions. 



On Reciprocal Plane Rectilinear Figures, 



DEFINITION. Two plane rectilinear figures are reciprocal when they consist 

 of an equal number of straight lines, so that corresponding lines in the two 

 figures are at right angles, and corresponding lines which meet in a point in 

 the one figure form a closed polygon in the other. 



[See pp. 169 and 188]. 



