AND DIAGRAMS OF FOKCES. 167 



in a determinate order of their sides, so arranged that, when we arrive at the 

 same side in naming the two polygons which it divides, we travel along it in 

 opposite directions. For instance, if pqr be one of the polygons, the others are 

 pbc, qca, rob. 



Note. It may be observed, that after drawing the lines p, q, r, b, c with 

 the parallel ruler, the line a was drawn by joining the points of concourse of 

 q, r and b, c ; but, since it represents the force in A, a is parallel to A. 

 Hence the following geometrical theorem : 



If the lines PQR, drawn from the angles of the triangle ABC, meet in a 

 point, then if pqr be a triangle with its corresponding sides parallel to P, Q, R, 

 and if a, b, c be drawn from its corresponding angles parallel to A, B, C, the 

 lines a, b, c will meet in a point. 



A geometrical proof of this is easily obtained by finding the centres of the 

 four circles circumscribing the triangles ABC, AQR, BRP, CPQ, and joining 

 the four centres thus found by six lines. 



These lines meet in the four centres, and are perpendicular to the six 

 lines, A, B, C; P, Q, R; but by turning them round 90 they become parallel 

 to the corresponding lines in the original figure. 



The diagram formed in this way is definite in size and position, but any 

 figure similar to it is a reciprocal diagram to the original figure. I have 

 explained the construction of this, the simplest diagram of forces, more at 

 length, as I wish to shew how, after the first line is drawn and its extremities 

 fixed on, every other line is drawn in a perfectly definite position by means of 

 the parallel ruler. 



In any complete diagram of forces, those forces which act at a given point 

 in the frame form a closed polygon. Hence, there will be as many closed 

 polygons in the diagram as there are points in the frame. Also, since each 

 piece of the frame acts with equal and opposite forces on the two points which 

 form its extremities, the force in the diagram will be a side of two different 

 polygons. These polygons might be drawn in any positions relatively to each 

 other ; but, in the diagrams here considered, they are placed so that each force 

 is represented by one line, which forms the boundary between the two polygons 

 to which it belongs. 



If we regard the polygons as surfaces, rather than as mere outlines, every 

 polygon will be bounded at every point of its outline by other polygons, so 



