AND DIAGRAMS OF FORCES. 



197 



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 

 proportional to the stresses in the two pieces, and it is positive or negative 

 according 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 

 parallelograms which can be formed with their sides parallel and equal to those 

 of the polygon. 



This is easily shewn by dividing the polygon into the different parallelo- 

 grams. 



On the Equilibrium of Stress in a Solid Body. 



Let PQR 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: 



