% Statics. 



example, that (friction and the gravity of the lever being left out 

 Fig. 74. of the question), the two weights p and q, would remain in equi- 

 librium in the inclined position represented, if, p being to q as Fq 

 to Fp, the surface of the lever merely rested upon the point F. 

 The fulcrum, in order that there may be an equilibrium in all posi- 

 tions of the lever, must have the effect of a pin passing through 

 it. In short, when we say, it is sufficient, that the resultant AF 

 of the two powers should pass through the fulcrum F, it is taken 

 for granted that the corresponding point F of the lever does not 

 admit of any motion ; for otherwise this condition is not sufficient. 

 Fig. 75. For example, if the lever BD were drawn by three forces jo, r/, r , 

 applied to the three chords Bp, D q, F r, there would not be an 

 equilibrium, if AF were the direction of the resultant of p and q, 

 although it should pass through the point F ; it would be further 

 64. necessary that the point of meeting A should be in r F. 



164. Since the two forces /?,</, in equilibrium by means of 

 Fig. 68, t- ne lever BFD^ must be in the inverse ratio of the perpendiculars 

 69< FL, FM ; that is, since it is necessary that/) should be to </, as 



FMto FL, it follows that/) X FL = q X FM\ in other words, 

 the moments of the two forces, taken with respect to the fulcrum, 

 64. or any other point in the direction AF, must be equal. 



165. As there cannot be a force without a tendency to mo- 

 tion, by the forces />, q, is to be understood the product of a cer- 

 tain mass by the velocities, that these forces would respectively 

 communicate to this mass, if it were free. Thus let m be a cer- 

 tain mass, and u the velocity that the force p, acting freely, is 

 capable of giving it ; also, let n be another mass, and v the veloc- 

 ity that the force q is capable of giving it ; in order that there 

 may be an equilibrium, the following proportion is necessary ; 

 namely, 



m X u : n X v :: FM : FL. 



166. Let 10 be the velocity produced by the force of gravity 

 _. ^ in an instant ; and let m, n, be two heavy bodies attached to two 



cords BI m, DKn, which passing over two round bodies 7, K, 

 transmit entirely to the lever BFD, according to any proposed 

 directions BI, DK, the action of gravity of these bodies ; we shall 

 have ic m, w n, as the measures of the forces with which these 



