AND DIAGRAMS OF FORCES. 189 



and c for <j). We have then for the equations of relation between the two 

 diagrams 



dz dz 



.(10). 



These equations are equivalent to the following definitions : 



Let z in the first diagram be given as a function of x and y, z will lie 

 on a surface of some kind. Let x , y be particular values of x and y, and 

 let z be the corresponding value of z. Draw a tangent plane to the surface at 

 the point x a , y , z , and from the point f=0, 7? = 0, = c; in the second 

 diagram draw a normal to this tangent plane. It will cut the plane at 

 the point %, 77 corresponding to xy, and the value of is equal and opposite 

 to the segment of the axis of z cut off by the tangent plane. The two 

 surfaces may be defined as reciprocally polar (in the ordinary sense) with respect 

 to the paraboloid of revolution 



(11), 



and the diagrams are the projections on the planes of xy and &) of points and 

 lines on these surfaces. 



If one of the surfaces is a plane-faced polyhedron, the other will also be a 

 plane-faced polyhedron, every face in the one corresponding to a point in the 

 other, and every edge in the one corresponding to the line joining the points 

 corresponding to the faces bounded by the edge. In the projected diagrams 

 every line is perpendicular to the corresponding line, and lines which meet in 

 a point in one figure form a closed polygon in the other. 



These are the conditions of reciprocity mentioned at p. 169, and it now 

 appears that if either of the diagrams is a projection of a plane-faced poly- 

 hedron, the other diagram can be drawn. If the first diagram cannot be a 

 projection of a plane-faced polyhedron, let it be a projection of a polyhedron 

 whose faces are polygons not in one plane. These faces must be conceived to 

 be filled up by surfaces, which are either curved or made up of different plane 

 portions. In the first case the polygon will correspond not to a point, but to 



