92 Statics. 



157. Let there be two forces or powers p, <?, applied at the 

 two points B, D, of the lever BFD, either immediately, or by 

 means of two cords, or two rods without mass, acting upon this 

 lever in the directions Bp, D q, and being in equilibrium. It is 

 proposed to determine the conditions of this equilibrium. 



As one of the two powers, q for example, cannot be in equi- 

 librium with the other except by means of the fulcrum F, it is 

 evident that the power q must produce two efforts, one of which 

 annihilates that of the power p, and the other is destroyed by 

 the fulcrum F, and consequently passes through this point. 



Let the lines p B, q D, representing the directions of the 

 powers be produced till they meet in some point A, and join AF. 

 43. The power q may be supposed to be applied at A, according to 

 A q ; then if AG represent the value or magnitude of this power, 

 and upon AG, as a diagonal, and in the directions AF, BAE, as 

 contiguous sides, we construct the parallelogram AHGE j JtE 

 will represent the effort made by the power q, according to this 

 line, and in a direction opposite to that of p ; and AH will be 

 that exerted against the fulcrum F. Indeed, although the point 

 A is not connected with the two points B, F, the force q is dis- 

 tributed in the same manner as if A were thus connected. For 

 it is evident, that if, without changing the forces or their direc- 

 tions, we connected the point A. with the three points B, jP, Z), 

 by means of three inflexible rods A.B, AF, AD, without mass, 

 this would not alter in any degree the supposed state of the 

 system or the manner in which the force q is exerted. Now in 

 this last case, the action of the force q would manifestly be com- 

 municate d in the manner we have mentioned ; therefore it would 

 be communicated in the same manner, according to the first 

 supposition. This being established, in order that there may be 

 an equilibrium, it is necessary that the force AE should not only 

 have a direction contrary to that of the force p, but that it should 

 also be equal to p. As to the force AH, in order that it may be 

 destroyed, it is sufficient that it be directed to the point F. Ac- 

 cordingly, if we designate the force exerted against the fulcrum 

 by p, we shall have 



q : p : p : : AG : AE : AH. 



