RECIPROCAL FIGURES, FRAMES, AND DIAGRAMS OF FORCES. 



31 



In this theorem a polygon is to be drawn for every point, whether the lines 

 of the frame meet or intersect, whether they are really jointed together, or 

 whether two pieces simply cross each other without mechanical connection. In 

 the latter case the polygon is a parallelogram, whose sides are parallel and pro- 

 portional to the stresses in the two pieces, and it is positive or negative accord- 

 ing as these stresses are of the same or of opposite signs. 



If three or more pieces intersect, it is manifestly the same whether they 

 intersect at one point or not, so that we have the following theorem : — 



The area of a polygon of an even number of sides, whose opposite sides are 

 equal and parallel, is equal to the sum of the areas of all the different parallelo- 

 grams which can be formed with their sides parallel and equal to those of the 

 polygon. 



This is easily shown by dividing the polygon into the different parallelograms. 



On the Equilibrium of Stress in a Solid Body. 



Let PQE be the longitudinal, and STU the tangential components of stress, 

 as indicated in the following table of stresses and strains, taken from Thomson 

 and Tait's "Natural Philosophy," p. 511, § 669 :— 



Components of the 



Planes, of which 



Relative Motion, or 



across which Force, 



is reckoned. 



Direction of 



Relative Motion 



or of Force. 



Strain. 



Stress. 



e 

 f 

 9 



a 

 b 

 c 



P 



Q 

 R 



S 

 T 

 U 



yz 

 zx 

 xy 



(yx 



\ zx 



\xy 

 f xz 



X 



y 



2' 



y 



z 

 z 



X 

 X 



y 



Then the equations of equilibrium of an element of the body are, by § 697 

 of that work, 



dP cW dT _ 



dx dy dz 



dU <iJQ 



dx dy 



f- + Y = 



dz 



dT dS dU „ A 

 dx dy dz 



(1)> 



