joi.] CONCURRENT FORCES. 55 



the figure, is equal and parallel to the resultant R, which is 

 therefore obtained by drawing through the point of intersection 

 of the forces a line equal and parallel to i 6. 



The graphical condition of equilibrium consists in the closing" 

 of the force polygon, that is, in the coincidence of its terminal 

 point (6) with its initial point (i). 



99. Analytically, -a systematic solution is obtained by resolv- 

 ing each force F into three components X, V, Z, along three 

 rectangular axes passing through the point of intersection of 

 the given forces. All components lying in the direction of the 

 same axis can then be added algebraically, and the whole system 

 of forces is found to be equivalent to three rectangular forces 

 2X, 2F, HZ, which, by the parallelogram law, can be combined 

 into a single resultant 



2F) 2 + (2Z) 2 



The angles a, /9, 7 made by this resultant with the axes are 

 given by the relations 



cos tt_cos/3__cos7_ i 

 ~~~' 



100. If the forces all lie in the same plane, only two axes are 

 required, and we have 



R = V(2.T) a + (2 F) 2 , tan **' 

 where 6 is the angle between the axis of X and R. 



101. The condition of equilibrium (Art. 97) R = o becomes, by 

 Art. 99, (E^) 2 +(^F) 2 +(^Z) 2 -o. As all terms in the left- 

 hand member are positive, their sum can vanish only when each 

 term is = o. The analytical conditions of the equilibrium of any 

 number of concurrent forces are therefore : 



=0, 2F=o, 2Z=0. 



