RECIPROCAL FIGURES, FRAMES, AND DIAGRAMS OF FORCES. 23 



The edge in the one diagram is perpendicular to the face in the other. 

 The distance of the plane from the origin represents the rate of increase of 

 the function along the edge. 



3. Every face in the one diagram corresponds to an edge in which as many 

 cells meet as there are angles in the face, that is, at least three. Every face 

 must belong to two, and only two cells, because the edge to which it corresponds 

 has two, and only two extremities. 



4. Every cell in the one diagram corresponds to a summit in the other. 

 Every face of the cell corresponds and is perpendicular to an edge having an 

 extremity in the summit. Since every cell must have four or more faces, every 

 summit must have four or more edges meeting there. 



Every edge of the cell corresponds to a face having an angle in the summit. 

 Since every cell has at least six edges, every summit must be the point of 

 concourse of at least six faces, which are the boundaries of cells. 



Every summit of the cell corresponds to a cell having a solid angle at the 

 summit. Since every cell has at least four summits, every summit must be the 

 meeting place of at least four cells. 



Mechanical Reciprocity of the Diagrams. 



If along each of the edges meeting in a summit forces are applied propor- 

 tional to the areas of the corresponding faces of the cell in the reciprocal 

 diagram, and in a direction which is always inward with respect to the cell, 

 then these forces will be in equilibrium at the summit. 



This is the "Polyhedron of Forces," and may be proved by hydrostatics. 



If the faces of the cell form a single closed surface which does not intersect 

 itself, it is easy to understand what is meant by the inside and outside of the 

 cell; but if the surface intersects itself, it is better to speak of the positive and 

 negative sides of the surface. A cell, or portion of a cell, bounded by a closed 

 surface, of which the positive side is inward, may be called a positive cell. If 

 the surface intersects itself, and encloses another portion of space with its 

 negative side inward, that portion of space forms a negative cell. If any portion 

 of space is surrounded by n sheets of the surface of the same cell with their 

 positive side inward, and by m sheets with their negative side inward, the space 

 enclosed in this way must be reckoned n — m times. 



In passing to a contiguous cell, we must suppose that its face in contact 

 with the first cell has its positive surface on the opposite side from that of the 

 first cell. In this way, by making the positive side of the surface continuous 

 throughout each cell, and by changing it when we pass to the next cell, we may 

 settle the positive and negative side of every face of every cell, the sign of 

 every face depending on which of the two cells it is considered for the moment 

 to belong to. 



