6 MR CLERK MAXWELL ON 



is represented by a single line, and in which the equilibrium of the forces meet- 

 ing at any point is expressed visibly by the corresponding lines in the other 

 figure forming a closed polygon. 



There are in this figure six lines, having four points of concourse, and form- 

 ing four triangles. To determine the direction of the force along a given line 

 at any point of concourse, we must make a point travel round the corresponding 

 polygon in the other figure in a direction which is positive with respect to that 

 polygon. For this purpose it is desirable to name the polygons in a determi- 

 nate 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 direc- 

 tions. For instance, if pqr be one of the polygons, the others are pbc, qca, rab. 



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 show 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 



