Leter. 99 



produced the directions Bp, E q, and taken from the point of 

 meeting A, the line AH to represent the power q, we decompose 

 this power into two others, one AG equal and directly opposite 

 to the power/?, and the other AC such as will admit of being in 

 equilibrium with the power r, by means of the fulcrum F, Ac- 

 cordingly, if the direction D r meet AC at the point /, we may 

 suppose the force AC applied at 7, according to the direction 

 ACIL ; the force AC or IL must therefore be capable of being 33. 

 decomposed into two others, one IK equal and directly opposite 

 to the power r, and the other IM directed against the fulcrum F. 

 Thus the force q produces the three effects AG, IK, IM, of 

 which the two first, being equal and directly opposite to the 

 forces/) and r, are destroyed, and the last, being directed against 

 the fixed point F, cannot but be destroyed also. Now, since all 

 the forces which act upon the lever, are p, q, r, or AG, IK, IM, 

 and AG, IK are destroyed, we conclude that IM is the resul- 

 tant of the three powers p, q, r, and that consequently the only 

 condition necessary to an equilibrium is, that this resultant 

 should pass through the fulcrum F. We see, therefore, that the 

 powers p, q, r, act upon the fulcrum as if they were immediately 

 applied to it according to directions parallel to those which they 

 actually have ; and this c6nclusion would hold true for any num- 

 ber whatever of powers, for we may always suppose one of the 

 powers to be in equilibrium with all the others by means of the 

 resistance of the fulcrum. 



1 72. Since F must be in one of the points of the resultant, it 

 must have the properties of which mention has already been 

 made ; that is, when several powers, exerted in the same plane, are 63 

 in equilibrium by means of a. lever of whatever figure, if from the 

 fulcrum we let fall perpendiculars upon the directions of these forces, 

 and multiply each force by the corresponding perpendicular, in other 

 words, if we take the moments of the forces with respect to the fulcrum, 

 the sum of the moments of the forces zvhich tend to turn the lever in 



about the fulcrum. The two powers p, q, being represented by AH, 

 and GA or AG' respectively, and being exerted in these directions, 

 are equivalent to A C or IL, the direction of which is opposed to r, 

 since it tends to turn the lever in the contrary direction ; and the 

 resultant of IL and IK', that is, of p, q, r, is IM. 



AH 



