2I3-] CONDITIONS OF EQUILIBRIUM. I2/ 



212. Doing the same thing for every force of the given sys- 

 tem and adding the components having the same direction, the 

 system will be found equivalent to the three rectangular forces 



SF, 



applied at O, together with the three couples 



whose vectors are at right angles. 



The three forces can now be compounded into a single 

 resultant 



whose direction is determined by the angles a, ft, 7. which it 

 makes with the axes Ox, Oy> Oz, 



^ Y 



C osa=- -, cos/3= , cos 7=- 

 K K K 



In the same way the three couples can be compounded into 

 a single resulting couple whose moment is 



213. Since R*, as well as // 2 , is thus found as the sum of 

 three squares, each of these quantities can vanish only if the 

 three squares composing it vanish separately. The conditions of 

 equilibrium of a rigid body (Art. 201) are therefore expressed 

 analytically by the following six equations : 



As the system of co-ordinates can be selected arbitrarily, the 

 meaning of the first three equations is that the sum of the com- 

 ponents of all the forces along any three lines not in the same 

 plane must vanish. The last three equations express that the 



