SIMPLE STRUCTURES 179 



These three conditions, ]P X= 0, ^ Y= 0, ^M= 0, are obviously 

 both necessary and sufficient to assure equilibrium. For if the 

 first two are satisfied, the system will be in equilibrium as regards 

 translation, and if ^M = 0, it will also be in equilibrium as 

 regards rotation ; and, furthermore, it will not be in equilibrium un- 

 less all three are satisfied. 



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. 



When a body is acted on by only three forces, lying in the same 

 plane, the conditions for equilibrium are that these three forces shall 

 meet in a point, and that one of them shall be equal and opposite 

 to the resultant of the other two. 



109. Equilibrium polygon. As explained above, the resultant of 

 any system of forces lying in the same plane may be found by means 

 of a vector force polygon, the resultant being the closing side of 

 the polygon formed on the given system of forces as adjacent sides. 

 Although this construction gives the magnitude and direction of 

 the resultant, it does not determine its position or its line of action. 

 The most convenient way to determine the line of action of the 

 resultant is to introduce into the given system two equal and oppo- 

 site forces of arbitrary amount and direction, such as P f and P" 

 (Fig. 118). Since P' and P" balance one another, they will not 

 affect the equilibrium of the given system. To find the line of 

 action of the resultant R, combine P' and P^ into a resultant R 1 

 acting along B'A', parallel to the corresponding ray OB of the force 

 polygon. Prolong A'B' until it intersects P 2 and then combine R l 

 and P 2 into a resultant R Z acting along C'B', parallel to the corre- 

 sponding ray 00 of the force polygon, etc. Proceed in this way 

 until the last partial resultant R is obtained. Then the resultant 



