AND DIAGRAMS OF FORCES. 169 



To every edge of the polyhedron will correspond the line which joins the 

 points corresponding to the two faces which meet in that edge. Each of these 

 lines is perpendicular to the projection of the other; for the perpendiculars 

 from the origin to the two faces, lie in a plane perpendicular to the edge in 

 which they meet, and the projection of the line corresponding to the edge is 

 the intersection of this plane with the plane of projection. Hence, the edge is 

 perpendicular to the projection of the corresponding line. The projection of the 

 edge is therefore perpendicular to the projection of the corresponding line, and 

 therefore to the corresponding line itself. In this way we may draw a diagram 

 on the plane of projection, every line of which is perpendicular to the corre- 

 sponding line in the original figure, and so that lines which meet in a point 

 in the one figure form a closed polygon in the other. 



If, in a system of rectangular co-ordinates, we make z = the plane of pro- 

 jection, and x = 0, y = Q, z = c the fixed point, then if the equation of a plane be 



z = Ax + By+C, 

 the co-ordinates of the corresponding point will be 



=cA, v = cB, =-C, 



and we may write the equation 



c(z + )=xt; + yr}. 



If we suppose 17, given as the co-ordinates of a point, then this equa- 

 tion, considering x, y, z as variable, is the equation of a plane corresponding 

 to the point. 



If we suppose x, y, z the co-ordinates of a point, and 77, as variable, 

 the equation will be that of a plane corresponding to that point. 



Hence, if a plane passes through the point xyz, the point corresponding 

 to this plane lies in the plane corresponding to the point xyz. 



These points and planes are reciprocally polar in the ordinary sense with 

 respect to the paraboloid of revolution 



We have thus arrived at a construction for reciprocal diagrams by con- 

 sidering each as a plane projection of a plane-sided polyhedron, these polyhedra 

 being reciprocal to one another, in the geometrical sense, with respect to a cer- 

 tain paraboloid of revolution. 



VOL. II. 22 



