and Diagrams of Forces. 



257 



Fis- 6. 



In fig. 6, let ABCDEbea 

 portion of a polygon bounded 

 by other polygons of which the 

 edges are P Q K S T, one or 

 more of these edges meeting 

 each angle of the polygon. 



In fig. VI., let abode be 

 lines parallel to ABCDE and 

 meeting in a point, and let these 

 be terminated by the lines 

 pqrst parallel to PQRST, 

 one or more of these lines com- 

 pleting each sector of fig. VII. 



In fig. 6 draw Y through the 

 intersections of A C and P Q, 

 and in fig. VI. draw y through 

 the intersections of a,p and c,q. 

 Then the figures of six lines 

 AB C P Q Y and a b cp qy will Fi g- VL 



be reciprocal, and y will be parallel to Y. Draw X parallel to x, 

 and through the intersections of T X and C E draw Z, and in 

 fig. VI. draw z through the intersections of c x and e t ; then 

 CDETXZ and cdetxz will be reciprocal, and Z will be pa- 

 rallel to z. Then through the intersections of AE and YZ 

 draw W, and through those of ay and e z draw w ; and since 

 A C E Y Z W and acey zw are reciprocal, W will be parallel 

 to w. 



By going round the remaining sides of the polygon ABCDE 

 in the same way, we should find by the intersections of lines 

 another point, the line joining which with the intersection of A E 

 would be parallel to w, and therefore we should have three points 

 in one line j namely, the intersection of Y and Z, the point de- 

 termined 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 pro- 

 jection of a plane-sided polyhedron are exactly the same. For 

 A being the intersection of the faces A P and A B, and C that of 

 B C and Q C, the intersection A C will be a point in the inter- 

 section of the faces A P and C Q. 



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

 that Y is the line of intersection of the faces A P and C Q. 



In the same way Z is the intersection of E T and C Q, so that 

 the intersection of Y and Z is a point in the intersection of A P 

 andET. 



Phil. Mag. S. 4. Vol. 27. No. 182. April 1864. S 



