524 ON RECIPROCAL FIGURES AND DIAGRAMS OF FORCES. 



The simplest case is that of five points in space with their ten connecting 

 lines, forming ten triangular faces enclosing five tetrahedrons. By joining the five 

 points which are the centres of the spheres circumscribing these five tetrahedrons, 

 we have a reciprocal figure of the kind described by Professor Rankine in the 

 J'ln'losophical Mayazine, February 1864; and forces proportional to the areas of 

 the triangles of one figure, if applied along the corresponding lines of connexion 

 of the other figure, will keep its points in equilibrium. 



In order to have perfect reciprocity between two figures, each figure must 

 be made up of a number of closed polyhedra having plane faces of separation, 

 and such that each face belongs to two and only two polyhedra, corresponding 

 to the extremities of the reciprocal line in the other figure. Every line in the 

 figure is the intersection of three or more plane faces, because the plane face in 

 the reciprocal figure is bounded by three or more straight lines. 



Let s be the number of points or summits, e the number of lines or edges, 



/ the number of faces, and c the number of polyhedra or cells. Then if about 



one of the summits in which polyhedra meet, and a- edges and 17 faces, we 



describe a polyhedral cell, it will have <f> faces and a- summits and rj edges, 



and we shall have 



s, the number of summits, will be decreased by one and increased by a- ; 

 c, the number of cells, will be increased by one ; 

 /, the number of faces, will be increased by $ ; 

 e, the number of edges, will be increased by 77 ; 



so that e + c (s+f) will be increased by 17+ 1 (<r + <f> 1), which is zero, or 

 this quantity is constant. Now in the figure of five points already discussed, 

 e=10, c = 5, *=5, f=lO; so that generally 



e + c = s+f, 

 in figures made up of cells in the way described. 



The condition of a reciprocal figure being indeterminate, determinate, or im- 

 possible except in particular cases, is 



e ^ 3s- 5. 



< 



This condition is sufficient to determine the possibility of finding a system of 

 forces along the edges which will keep the summits in equilibrium ; but it is 



