162 RECIPROCAL FIGURES, FRAMES, 



by straight lines, and we may indicate by numbers attached to these lines the 

 tensions or pressures in the corresponding pieces of the frame. 



The diagram thus formed indicates the state of the frame in a way which 

 is geometrical as regards the position and direction of the forces, but arith- 

 metical as regards their magnitude. 



But, by assuming that a line of a certain length shall represent a force 

 of a certain magnitude, we may represent every force completely by a line. 

 This is done in Elementary Statics, where we are told to draw a line from 

 the point of application of the force in the direction in which the force acts, 

 and to cut off as many units of length from the line as there are units of 

 force in the force, and finally to mark the end of the line with an arrow- 

 head, to shew that it is a force and not a piece of the frame, and that it 

 acts in that direction and not the opposite. 



By proceeding in this way, we should get a system of arrow-headed forces 

 superposed on the skeleton of the frame, two equal and opposite arrows for 

 every piece of the frame. 



To test the equilibrium of these forces at any point of concourse, we 

 should proceed by the construction of the parallelogram of forces, beginning 

 with two of the forces acting at the point, completing the parallelogram, and 

 drawing the diagonal, and combining this with the third force in the same way, 

 till, when all the forces had been combined, the resultant disappeared. We 

 should thus have to draw three new lines, one of which is an arrow, in taking 

 in each force after the first, leaving at last not only a great number of useless 

 lines, but a number of new arrows, not belonging to the system of forces, and 

 only confusing to any one wishing to verify the process. 



To simplify this process, we are told to construct the Polygon of Forces, 

 by drawing in succession lines parallel and proportional to the different forces, 

 each line beginning at the extremity of the last. If the forces acting at the 

 point are in equilibrium, the polygon formed in this way will be a closed one. 



Here we have for the first time a true Diagram of Forces, in which every 

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

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

 to examine whether the polygon is closed or not. To secure this advantage, 

 however, we have given up the attempt to indicate the position of the force, 

 for the sides of the polygon do not pass through one point as the forces do. 

 We must, therefore, give up the plan of representing the frame and its forces 



