184 RECIPROCAL FIGURES, FRAMES, 



or the value of F at the origin is equal and opposite to the value of <f> at 

 the point , 17, {. 



If the rate of variation of F is nowhere infinite, the co-ordinates , 77, , 

 of the second diagram must be everywhere finite, and vice versa. Beyond the 

 limits of the second diagram the values of x, y, z, in terms of , 77, , must 

 he impossible, and therefore the value of ^ is also impossible. Within the 

 limits of the second diagram, the function < has an even number of values at 

 every point, corresponding to an even number of points in the first diagram, 

 which correspond to a single point in the second. 



To find these points in the first diagram, let p be the vector of a given 

 point in the second diagram, and let surfaces be drawn in the first diagram 

 for which F is constant, and let points be found in each of these surfaces at 

 which the tangent plane is perpendicular to p, these points will form one or 

 more curves, which must be either closed or infinite, and the points on these 

 curves correspond to the points in the second diagram which lie in the 

 direction of the vector p. If p be the perpendicular from a point in the first 

 diagram on a plane through the origin perpendicular to p, then all those points 



on these curves at which ~r- = p correspond to the given point in the second 

 diagram. Now, since this point is within the second diagram, there are values 



of p both greater and less than the given one ; and therefore -y- is neither an 



absolute maximum nor an absolute minimum value. Hence there are in general 

 an even number of points on the curve or curves which correspond to the 

 given point. Some of these points may coincide, but at least two of them 

 must be different, unless the given point is at the limit of the second diagram. 



Let us now consider the two reciprocal diagrams with their functions, and 

 ascertain in what the geometrical nature of their reciprocity consists. 



(1) Let the first diagram be simply the point P u (x u y u z,), at which 

 F=F U then in the other diagram 



(6), 



or a point in one diagram is reciprocal to a space in the other, in which the 

 function ^ is a linear function of the co-ordinates. 





