and Diagrams of Forces. 



255 



In figures 4 and IV. the condition that the number of polygons 

 is equal to the number of points is not fulfilled. In fig. 4 there 



Fig. IV. 



Fig. 4. 



V 





V> N v 





V\\ 







\ \ ^ 





\ \ n 



V \ 



s 







\ ' 



>v 



\ 



\ ■'" ' 



\ 



\ '' ! 



\ 



V .' 



are five points and six triangles ; in fig. IV. there are six points, 

 two triangles, and three quadrilaterals. Hence if fig. 4 is given, 

 fig. IV. is indeterminate to the extent of one variable, besides 

 the elements of scale and position. In fact when we have drawn 

 ABC and indicated the directions of P, Q, R, we may fix on 

 any point of P as one of the angles of X Y Z and complete the 

 triangle XYZ. The size of XYZ is therefore indeterminate. 

 Conversely, if fig. IV. is given, fig. 4 cannot be constructed unless 

 one condition be fulfilled. That condition is that P, Q, and R 

 meet in a point. When this is fulfilled, it follows by geometry 

 that the points of concourse of A and X, B and Y, and C and Z 

 lie in one straight line W, which is parallel to w in fig. 4. The 

 condition may also be expressed by saying that fig. IV. must be a 

 perspective projection of a polyhedron whose quadrilateral faces 

 are planes. The planes of these faces intersect at the concourse 

 of P, Q, R, and those of the triangular faces intersect in the 

 line W. 



Figs. 5 and V. represent another case of the same kind. In 

 fig. 5 we have six points and eight triangles j fig. V. is therefore 



