48 WHICH EQUATIONS TO USE. Art. 41 



1. Two resolution and one moment equation. 



2. One resolution and two moment equations 1 . 



3. Three moment equations 2 . 



Since there must be equilibrium for any center of moments, 

 it is plain that an infinite number of moment equations may be 

 written. That any three of them or any two of them are inde- 

 pendent is not so apparent from what precedes. If the moment 

 of the resultant is zero for any point, it can not be a couple, be- 

 cause the moment of a couple is the same for every point. If the 

 sum of the moments is zero for two different points, therefore, the 

 line of action of the resultant passes through these points; this 

 fixes its direction ; it remains to impose another condition making 

 it equal to zero. If the moment of the resultant is zero for any 

 point not in its line of action, the resultant must be zero; or if it 

 has no component in any one direction it is zero; it has no com- 

 ponent in one direction if the forces are in equilibrium and the 

 components are inclined to it or parallel to it. 



Graphically combinations 2 and 3 are of no importance, 

 because they involve more labor than the first combination. 



It is plain that, with the exceptions noted, any one of the 

 three combinations of the three equations of equilibrium may be 

 used ; it is simply a matter of convenience. The applications will 

 be illustrated under the different methods. 



When the forces are concurrent, there are only two inde- 

 pendent equations, as follows : 



1. Two resolution equations. 



2. One resolution and one moment equation. 



3. Two moment equations. 



These equations have various advantages and disadvantages 

 in their practical use. The resolution equations involve the deter- 

 mination of sines and cosines, while the moment equations involve 

 the calculation of moment arms. Except in special cases, tfce 

 unknowns are found by elimination between several resolution 

 equations, while a moment equation may always be written con- 

 taining but one unknown, if the center of moments is chosen upon 

 the line of action of an unknown force, or at the intersection of 

 two unknown forces. The moment equations are the simplest for 



centers of moments must not be in a line perpendicular to the 

 components. 



2 The centers of moments must not be in the same straight line. 



