],,.; RECIPROCAL FIGURES, FRAMES, 



that the whole assemblage of polygons will form a continuous surface, which 

 muat either be an infinite surface or a closed surface. 



The diagram cannot be infinite, because it is made up of a finite number 

 of finite lines representing finite forces. It must, therefore, be a closed surface 

 returning on itself, in such a way that every point in the plane of the diagram 

 either does not belong to the diagram at all, or belongs to an even number 

 of sheets of the diagram. 



Any system of polygons, which are in contact with each other externally, 

 may be regarded as a sheet of the diagram. When two polygons are on the 

 same side of the line, which is common to them, that line forms part of the 

 common boundary of two sheets of the diagram. If we reckon those areas 

 positive, the boundary of which is traced in the direction of positive rotation 

 round the area, then all the polygons in each sheet will be of the same sign as 

 the sheet, but those sheets which have a common boundary will be of opposite 

 sign. At every point in the diagram there will be the same number of positive 

 as of negative sheets, and the whole area of the positive sheets will be equal to 

 that of the negative sheets. 



The diagram, therefore, may be considered as a plane projection of a closed 

 polyhedron, the faces of the polyhedron being surfaces bounded by rectilinear 

 polygons, which may or may not, as far as we yet know, lie each in one plane. 



Let us next consider the plane projection of a given closed polyhedron. 

 If any of the faces of this polyhedron are not plane, we may, by drawing 

 additional lines, substitute for that face a system of triangles, each of which is 

 necessarily in a plane. We may, therefore, consider the polyhedron as bounded 

 by plane faces. Every angular point of this polyhedron will be defined by its 

 projection on the plane and its height above it. 



Let us now take a fixed point, which we shall call the origin, and draw 

 from it a perpendicular to the plane. We shall call this line the axis. If we 

 then draw from the origin a line perpendicular to one of the faces of the poly- 

 hedron, it will cut the plane at a point which may be said to correspond to the 

 projection of that face. From this point draw a line perpendicular to the plane, 

 and take on this line a point whose distance from the plane is equal to that 

 of the intersection of the axis with the face of the polyhedron produced, but on 

 the other side of the plane. This point in space will correspond to the face of 

 the polyhedron. By repeating this process for every face of the polyhedron, we 

 shall find for every face a corresponding point with its projection on the plane. 



