25 2 Prof. Maxwell on Reciprocal Figures 



General Relation between the Numbers of Points, Lines, and 

 Polygons in Reciprocal Figures. 



The effect of drawing a line, one of whose extremities is a 

 point connected with the system of lines already drawn, is either 

 to introduce one new point into the system, or to complete one new 

 polygon, or to divide a polygon into two parts, according as it is 

 drawn to an isolated point, or a point already connected with the 

 system. Hence the sum of points and polygons in the system 

 is increased by one for every new line. But the simplest figure 

 consists of four points, four polygons, and six lines. Hence the 

 sum of the points and polygons must always exceed the number 

 of lines by two. 



Note. — This is the same relation which connects the numbers 

 of summits, faces, and edges of polyhedra. 



Conditions of indeterminateness and impossibility in drawing 

 reciprocal Diagrams. 



Taking any line parallel to one of the lines of the figure for a 

 base, every new point is to be determined by the intersection of 

 two new lines. Calling s the number of points or summits, e 

 the number of lines or edges, and / the number of polygons or 

 faces, the assumption of the first line determines two points, and 

 the remaining s—2 points are determined by 2(s — 2) lines. 

 Hence if e = 2s-3, 



every point may be determined. If e be less, the form of the 

 figure will be in some respects indeterminate ; and if e be greater, 

 the construction of the figure will be impossible, unless certain 

 conditions among the directions of the lines are fulfilled. 



These are the conditions of drawing any diagram in which the 

 directions of the lines are arbitrarily given ; but when one dia- 

 gram is already drawn in which e is greater than 25—3, the 

 directions of the lines will not be altogether arbitrary, but will 

 be subject to e— (2s — 3) conditions. 



Now if e' } s J ,f' be the values of e, s, and/ in the reciprocal 

 diagram 



e = e>, s=f, f=s\ 

 e = s+f-2, e' = s<+f , -2. 

 Hence if s=f, e = 2i—2; and there will be one condition con- 

 necting the directions of the lines of the original diagram, and 

 this condition will ensure the possibility of constructing the reci- 

 procal diagram. If 



s >/, e>2s— 2, and e'<2s , — 2; 

 so that the construction of the reciprocal diagram will be pos- 

 sible, but indeterminate to the extent of s—f variables. 



