CHAP. I.] COMMON POINT OF APPLICATION. 3 



with the point of beginning, there is, of course, no resultant, 

 and the forces themselves are in equilibrium. 



4. The polygon formed by the successive lay'ng off of the 

 lines parallel and equal to the forces, we call the "force poly- 

 gon" Hence we have the following principles established : 



If any number of forces having a common point of appli- 

 cation and lying in the same plane, are in equilibrium , the 

 "force polygon" is closed. 



If the "force polygon " is not closed, the forces themselves 

 are not in equilibrium, and the line necessary to close it gives 

 the resultant in intensity and direction. 



This resultant, if considered as acting in the direction ob- 

 tained l>y following round the " force polygon" with the forces, 

 will produce equilibrium acting in the opposite direction, it 

 replaces the forces. 



The resultant thus found in intensity and direction can be 

 inserted in the force diagram at the common point of applica- 

 tion. 



5. Thus, required the position, intensity, and direction of the 

 resultant of the forces P 1? P 2 , P 3 , P 4 , P 5 . 



These forces are given in position, direction, and intensity 

 by the force diagram, Fig. 3 (a). The resultant of all these 

 forces must have of course the same point of application A as 

 the forces themselves it remains to find then its relative posi- 

 tion and the direction of its action, so that we may properly 

 insert it in the force diagram. 



We have simply to draw the force polygon, Fig. 3, (&) by lay- 

 ing off successively O P l5 P! P 2 , etc., equal, parallel, and in the 

 same direction as the forces P l5 P 2 , etc., as given by Fig. 3 (a). 

 Then the line P 5 O necessary to close the force polygon gives 

 the intensity of the resultant, and in order to replace P w it 

 must act in the direction from O to P 5 ; i.e., contrary to the 

 order of the forces. If then in Fig. 3 (a) we draw A R^ equal 

 and parallel to O P 5 , we have the resultant applied at the com- 

 mon point of application A, and given in position, intensity 

 and direction. 



Moreover, it is evident that any diagonal of the force poly- 

 gon aa Rg^ [Fig. 3 ()] is the resultant of P^, and acting in the 

 direction from P 4 to P 2 , it holds P M in equilibrium. But it is 

 also the resultant of P M P 2 , P 5 , and R w , and acting in the same 

 direction as before, it replaces these forces. The force polygon 



