516 ON RECIPROCAL FIGURES AND DIAGRAMS OF FORCES. 



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 f the number of polygons or faces, the assumption of the first line deter- 

 mines 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 diagram is already drawn in 

 which e is greater than 2* 3, the directions of the lines will not be altogether 

 arbitrary, but will be subject to e (2s 3) conditions. 



Now if e', s', f be the values of e, s, and f in the reciprocal diagram 



e = e', *=/', f=s', 

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



Hence if s =f, e = 2i 2; and there will be one condition connecting the 

 directions of the lines of the original diagram, and this condition will ensure 

 the possibility of constructing the reciprocal diagram. If 



s >/, e > 2s - 2, and e' < 2s' - 2 ; 



so that the construction of the reciprocal diagram will be possible, but inde- 

 terminate to the extent of s f variables. 



If s<f, the construction of the reciprocal diagram will be impossible unless 

 (a/) conditions be fulfilled in the original diagram. 



