54 
Proceedings of the Royal Society 
the action of pressures and tensions in its several pieces, the other 
figure represents a system of forces which would keep the frame in 
equilibrium ; and, if the known data are sufficient to determine 
these forces, the reciprocal figure may be drawn so as to represent, 
on a selected scale, the actual values of all these forces. 
In this way a practical method of determining the tensions and 
pressures in structures has been developed. The “polygon of 
forces ’’has been long known. The application to polygonal frames, 
with a system of forces acting on the angles, and to several other 
cases, was made by Professor Rankine in his Applied Mechanics. 
Mr W. P. Taylor, a practical draughtsman, has independently 
worked out more extensive applications of the method. Starting 
from Professor Rankine’s examples, I taught the method to the 
class of Applied Mechanics in King’s College, London, and published 
a short account of it in the “Philosophical Magazine” for April 
1864. Professor Pleeming Jenkin, in a paper recently presented 
to the Society, has fully explained the application of the method to 
the most important cases occurring in practice, and I believe that 
it has been found to have three important practical advantages. 
It is easily taught to any person who can use a ruler and scale. 
It is quite sufficiently accurate for all ordinary calculations, and is 
much more rapid than the trigonometrical method. When the 
figure is drawn the whole process remains visible, so that the 
accuracy of the drawing of any single line can be afterwards tested ; 
and if any mistake has been made, the figure cannot be completed. 
Hence the verification of the process is much easier than that ol‘ 
a long series of arithmetical operations, including the use of 
trigonometric tables. 
In the present paper I have endeavoured to develope the idea of 
reciprocal figures, to show its connection with the idea of reciprocal 
polars as given in pure mathematics, and to extend it to figures in 
three dimensions, and to cases in which the stresses, instead of 
being along certain lines only, are distributed continuously through- 
out the interior of a solid body. In making this extension of the 
theory of reciprocal figures, I have been led to see the connection 
of this theory with that of the very important function introduced 
into the theory of stress in two dimensions by Mr Airy, in his paper 
“On the Strains in the Interior of Beams” (Phil. Trans. 1863). 
