ON RECIPROCAL FIGURES AND DIAGRAMS OF FORCES. 521 



and cdetxz will be reciprocal, and Z will be parallel to z. Then through the 

 intersections of AE and YZ draw W, and through those of ay and ez draw 

 w; and since ACEYZW and aceyzw are reciprocal, W will be parallel to w. 



Fig. 6. 



Fig. VI. 



By going round the remaining sides of the polygon ABODE in the same 

 way, we should find by the intersections of lines another point, the line joining 

 which with the intersection of AE would be parallel to w, and therefore we 

 should have three points in one line ; namely, the intersection of Y and Z, 

 the point determined by a similar process carried on on the other part of the 

 circumference of the polygon, and the intersection of A and E ; and we should 

 find similar conditions for every pair of sides of every polygon. 



Now the conditions of the figure 6 being a perspective projection of a 

 plane-sided polyhedron are exactly the same. For A being the intersection of 

 the faces AP and AB, and C that of BC and QC, the intersection AC will 

 be a point in the intersection of the faces AP and CQ. 



Similarly the intersection PQ will be another point in it, so that Y is the 

 line of intersection of the faces AP and CQ. 



In the same way Z is the intersection of ET and CQ, so that the inter- 

 section of Y and Z is a point in the intersection of AP and ET. 



Another such point can be determined by going round the remaining sides 

 of the polygon ; and these two points, together with the intersections of the 

 lines AE, must all be in one straight line, namely, the intersection of the faces 

 AP and ET. 



Hence the conditions of the possibility of reciprocity in plane figures are 

 the same as those of each figure being the perspective projection of a plaiie- 

 sided polyhedron. When the number of points is in every part of the figure 

 equal to or less than the number of polygons, this condition is fulfilled of 

 itself. When the number of points exceeds the number of polygons, there will 



VOL. i. 66 



