Principle of Virtual Velocities. 133 



and as this ratio remains the same, however small the motion 

 impressed upon the lever, it holds true, when u : v : : q : p, 

 which gives exactly 



q : p : : FL : FM. 



In the case of an equilibrium, therefore, the two forces are to 

 each other in the inverse ratio of the perpendiculars let fall upon 

 their directions, as already determined by a different method. 



230. If we had taken the extremities B, D, of the lever for 

 the points of application of the forces p, q, the directions of the 

 forces would no longer be tangents to the arcs described by these 

 points. We should therefore have to project these arcs upon 

 the straight lines Bp, D q, then to take the ratio of these projec- 

 tions, and seek the limit of this ratio. 



On the supposition of motion, the angles DFD', BFB', describ- 

 ed by the arms FB,FD, are equal; and the arcs BB', DD', des- 

 cribed by B, D, about the point F, as a centre, are to each other 

 as the radii FB, FD. This ratio continues the same when the 

 arcs become infinitely small, so that we have constantly 



FB : BB / :: FD : DD / . 



From the points B', D, let fall the perpendiculars B'A, DC, upon 

 the directions of the forces p, q ; and we shall have 



u = BA, and -o = DC. 



Also from F, let fall the perpendiculars FM, FL, upon the direc- 

 tions of the forces. By considering the infinitely small arcs 

 BB', DD', as straight lines, perpendicular respectively to the Geom. 

 radii FB, FD, the triangles DD'C, FMD, are similar, as also the 209 - 

 triangles BB'A, FLB-, whence 



FB : FL : : BB' : BA = J^L x FL, 

 and 



FD : FM : : DD' : DC = ^ x FM; 

 accordingly we have, by substitution, 



D f)f T-) TV 



u = -JjfL- x FL, and v = -~L x FM, 



