8 MR CLERK MAXWELL ON 



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 

 corresponding 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=0 the plane of pro- 

 jection, and x = y = 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 f = - c , 



and we may write the equation 



<*+£) = a£ + yr, . 



If we suppose £, ??, £ given as the co-ordinates of a point, then this equation, 

 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 £, -q, £ 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 



2cz = o? + y 2 . 



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

 ing 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. 



Each of the diagrams must fulfil the conditions of being a plane projection 

 of a plane-sided polyhedron, for if any of the sides of the polyhedron of which 

 it is the projection are not plane, there will be as many points corresponding to 

 that side as there are different planes passing through three points of the side, 

 and the other diagram will be indefinite. 



Belation between the Number of Edges, Summits, and Faces of Polyhedra. 



It is manifest that after a closed surface has been divided into separate faces 

 by lines drawn upon it, every new line drawn from a point in the system, either 

 introduces one new point into the system, or divides a face into two parts, 



