AKCHES AND ARCHED EIBS 



161 



The resultant of a system of non-concurrent forces lying in the 

 same plane, that is to say, forces whose lines of action do not all pass 

 through the same point, is found by means of a force polygon as 

 explained above. In this case, however, the closing of the force 

 polygon is not a sufficient condition for equilibrium, for the given 

 system may reduce to a pair of equal and opposite forces acting in 

 parallel directions, called a couple, which would tend to produce rota- 

 tion of the body on which they act. For non-concurrent forces, there- 

 fore, the necessary and sufficient conditions for equilibrium are first, 

 the resultant of the given system must be zero, and second, the sum of 

 the moments of the forces about 

 any point must be zero. 



Suppose that the force polygon 

 corresponding to any given system 

 of forces is projected upon two 

 perpendicular lines, say a vertical 

 and a horizontal line. Then since 

 the sum of the projections upon 

 any line of all the sides but one 

 of a polygon is equal to the pro- 

 jection of this closing side upon 

 the given line, the sum of the horizontal projections of any system 

 of forces is equal to the horizontal projection of their resultant, and 

 the sum of their vertical projections is equal to the vertical projection 

 of their resultant (Fig. 1 16). 



The conditions for equilibrium of a system of forces lying in the 

 same plane may then be reduced to the following convenient form. 



1. For equilibrium against translation, 



^horizontal components = O, 

 vertical components = O. 



2. For equilibrium against rotation, 



^moments about any point = O. 



Tf the forces are concurrent, rotation cannot occur, and the first 

 condition alone is sufficient to assure equilibrium. In order that 



