104 STATICS. [174. 



in magnitude and direction. In other words, when the forces 

 are not parallel, they must be added geometrically, and not 

 algebraically. 



The construction of the funicular polygon and its properties 

 are the same as for parallel forces. 



If the force polygon does not close, the given system is 

 equivalent to a single resultant represented in magnitude, direc- 

 tion, and sense by the closing line ; its position is obtained from 

 the funicular polygon whose initial and final lines must inter- 

 sect on the resultant. 



If, however, the force polygon closes, the system may be 

 equivalent to a couple, or it may be in equilibrium. The dis- 

 tinction between these two cases is indicated by the funicular 

 polygon. If the initial and final lines of this polygon coincide, 

 the system is in equilibrium ; if they are merely parallel, these 

 lines are the directions of the forces of the couple to which the 

 whole system reduces. The magnitude and sense of the forces 

 of the resulting couple are obtained from the force polygon. 



174. Thus it follows from the graphical as well as from the 

 analytical method that a plane system may be equivalent to a 

 single force, or to a couple, or to zero. In the first case, the force 

 polygon does not close, and the initial and final sides of the 

 funicular polygon intersect at a finite distance. In the second 

 case, the force polygon closes, and the initial and final lines of 

 the funicular polygon are parallel. In the third case, the force 

 polygon closes, and the initial and final sides of the funicular 

 polygon coincide. 



The graphical conditions of equilibrium of a plane system are, 

 therefore, two: (i) the force polygon must close; (2) the funi- 

 cular polygon must have its initial and final sides coincident. 



175. To every vertex of the force polygon corresponds a side 

 of the funicular polygon, and vice versa. The force polygon is 

 said to close if the last vertex coincides with the first ; similarly, 

 the funicular polygon might be said to close when its last side 



