186 RECIPROCAL FIGURES, FRAMES, 



The bounding surfaces of these parts of space must be composed of planes. 

 For let the linear functions of xyz in contiguous portions of space be 



then at the bounding surface, where F 1 = F t 



(a,-a.)a: + (A-A)y + (y,-y,)z = *,-6 .................. (9), 



and this is the equation of a plane. 



Hence the portion of space in which any particular form of the value of F 

 holds good must be a polyhedron or cell bounded by plane faces, and therefore 

 having straight edges meeting in a number of points or summits. 



Every face is the boundary of two cells, every edge belongs to three or 

 more cells, and to two faces of each cell. 



Every summit belongs to at least four cells, to at least three faces of each 

 cell, and to two edges of each face. 



The whole space occupied by the diagram is divided into cells in two 

 different ways, so that every point in it belongs to two different cells, and 

 has two values of F and its derivatives. 



The reciprocal diagram is made up of cells in the same way, and tlie 

 reciprocity of the two diagrams may be thus stated : 



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



The radius vector of the summit represents the rate of increase of the 

 function within the cell, both in direction and magnitude. 



The value of the function at the summit is equal and opposite to the 

 value which the function in the cell would have if it were continued under the 

 same algebraical form to the origin. 



2. Every edge in the one diagram corresponds to a plane face in the 

 other, which is the face of contact of the two cells corresponding to the two 

 extremities of the edge. 



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. 



