FORCES IN TITE SAME PLANE. [CHAP. I 



thus shows that the force which replaces P t , P 2 , P 5 , and R w , at 

 the same time holds P 3 and P 4 in equilibrium, just as it should 

 do. 



If, on the other hand, we had originally only P t , P 2 , R^, P 5 , 

 and R w forming a system of forces in equilibrium, we could 

 decompose Rg^ into two components by simply assuming any 

 point as P 8 [Fig. 3 (J)] and drawing P 3 P 4 , P 3 P 2 . Then follow- 

 ing round this new polygon in the direction of the forces, or, 

 what amounts to the same thing, taking the direction of the 

 components P 8 P 4 , opposed to the direction of R^ for equilibri- 

 um, we obtain the direction of action of P 8 and P 4 as shown by 

 the arrows in Fig. 3 (b). These forces inserted in Fig. 3 (a), in 

 the place of Rg 4 and in these directions, will not disturb the 

 equilibrium. 



Hence, any diagonal in the force polygon, is the resultant 

 of the forces on either side, holding in equilibrium those on 

 one side and replacing those on the other, according to the 

 direction in which it is conceived to act. 



Also, any force or number of forces may be decomposed into 

 two others in any desired direction, by choosing a suitable 

 point in the plane of the force polygon and drawing lines 

 from this point to the beginning and end of the force or force 

 polygon. 



6. It mailers not in what Order we lay off the Forces in 

 the Construction of the force Polygon. Thus, in Fig. 1, 

 whether we draw from the end of P 2 the line P 2 R^ equal and 

 parallel to P! or from the end of P t the line P t R^ equal and 

 parallel to P 2 , in either case we obtain the same resultant and 

 the same direction for the resultant. But by a similar change 

 of two and two, we can obtain any order we please. For exam- 

 ple, we lay off in Fig. 3 (c) the same forces in the order P s P a 

 PU P 5 P 4 , and obtain precisely the same resultant, in the same 

 direction as before. For, the resultant of P 8 and P 2 must be 

 the same as that of P 2 and P 3 in the first case. The resultant 

 of Rg.,, and P t must then be the same in both polygons, and so 

 on. 



Generally, then, no matter what the order in which the 

 forces are laid off, the line necessary to close the force polygon 

 is the resultant of the forces, and the diagonals of the force 

 polygon give us the resultants of the forces on either side. 



By assuming a point at pleasure, and drawing lines from this 



