OF THE EQUILIBRIUM OF A SYSTEM. 177 



entered very fully and clearly into the history of this pro- 

 position. 



305. Theorem. If a system of bodies be 

 in equilibrium, the sum of the products of the 

 forces, acting on the several bodies, into the 

 infinitely small variations of their places, in the 

 directions of the forces, the variations being so 

 taken as to be subjected to the conditions of 

 the system, must be equal to nothing. Or, if 

 p be the force acting on each body, and dfthe 

 variation of the place of the body in its direc- 

 tion, 0=xp^f; which is the Law of virtual velo- 

 cities. 



Let us first suppose two heavy bodies, m and m\ fixed 

 to the extremities of a horizontal line, supposed to be in- 

 flexible and without weight, being at liberty to turn round 

 a fixed point within its length. In order to conceive the 

 action of these bodies on each other when they are in equi- 

 librium, we must suppose the right line to be infinitely little 

 bent at the fixed point, so as to be formed of two right 

 lines, making at that point an angle which differs but infi- 

 nitely little from two right angles ; and this difference we 

 may call «. Let f and f be the distances of m and m^ 

 from the fixed point ; if we decompose the weight of m 

 into two parts, the one acting on the fixed point, in the 

 direction of the bent line, the other directed towards 7nf, 



this last will be , , mg being the weight of the 



body : [for since AB : sin ADBziDB : sin DAC, (P. 175) 



