RECIPEOCAL FIGURES, FRAMES, AND DIAGRAMS OF FORCES. 7 



polygon will be bounded at every point of its outline by other polygons, so 

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

 must 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 polyhedron, 

 it will cut the plane at a point which may be said to correspond to the projec- 

 tion 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. 



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 



