22 MR CLERK MAXWELL ON 



(4.) Let there be a fourth point P 4 for which F = F 4 . 



The reciprocal of the four points is a single point, and the line drawn from 

 the origin to this point represents, in direction and magnitude, the rate of 

 greatest increase of F, supposing F such a linear function of xyz that its values 

 at the four points are those given. The value of <f> at this point is that of F at 

 the origin. 



Let us next suppose that the value of F is continuous, that is, that F does 

 not vary by a finite quantity when the co-ordinates vary by infinitesimal 

 quantities, but that the form of the function F is discontinuous, being a 

 different linear function of xyz in different parts of space, bounded by definite 

 surfaces. 



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 



Fj = a x x + fay + 7l s - fa 



F 2 = a r >- + j3 2 y + y 2 z - fa, 



then at the bounding surface, where F x = F 2 



{a x -a 2 )x + (P x -fa)*f + (7i-7-2)- = 01— 4>i ■ ( 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 the 

 reciprocity of the two diagrams may be thus stated : — 



1. Every summit in one diagram corresj3onds to a cell in the other. 



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

 tion within the cell, both in direction and magnitude. 



The value of the function at the summit is equal and ojDposite 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. 



