AND DIAGRAMS OF FORCES. 187 



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 pro- 

 portional 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 inter- 

 sect 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 



242 



