BOW'S METHOD OF DRAWING DIAGRAMS IN GRAPHICAL STATICS. 493 



units of force in the force. The end of the line is marked by an arrow-head 

 to shew in which direction the force acts. 



According to this method each force is drawn in its proper position in 

 the diagram which represents the configuration of the system. Such a diagram 

 might be useful as a record of the results of calculation of the magnitude of 

 the forces, but it would be of no use in enabling us to test the correctness 

 of the calculation. It would be of less use than the diagram in which the 

 magnitudes of the forces were indicated by numbers. . , 



But we have a geometrical method of testing the equilibrium of any set 

 of forces acting at a point by drawing in series a set of lines parallel and 

 proportional to these forces. If these lines form a closed polygon the forces 

 are in equilibrium. We might thus form a set of polygons of forces, one for 

 each joint of the frame. But in so doing we give up the principle of always 

 drawing the line representing a force from its point of application, for all the 

 sides of a polygon cannot pass through the same point as the forces do. 



We also represent every stress twice over, for it appears as a side of both 

 the polygons corresponding to the two joints between which it acts. 



But if we can arrange the polygons in such a way that the sides of any 

 two polygons which represent the same force coincide with each other, we may 

 form a diagram in which every stress is represented in direction and magnitude, 

 though not in position, by a single line, which is the common boundary of the 

 two polygons which represent the points of concourse of the pieces of the frame. 



Here we have a pure diagram of forces, in which no attempt is made to 

 represent the configuration of the material system, and in which every force 

 is not only represented in direction and magnitude by a straight line, but the 

 equilibrium of the forces is manifest by inspection, for we have only to examine 

 whether each polygon is closed or not. 



The relations between the diagram of the frame and the diagram of stress 

 are as follows : 



To every piece in the frame corresponds a line in the diagram of stress 

 which represents in magnitude and direction the stress acting on that piece. 



To every joint of the frame corresponds a closed polygon in the diagram, 

 and the forces acting at that joint are represented by the sides of the polygon 

 taken in a certain cyclical order. The cyclical order of the sides of two adjacent 

 polygons is such that their common side is traced in opposite directions in 

 going round the two polygons. 



