132 Statics. 



directly from the ratio of the spaces which the points of appli- 

 cation of the two forces tend to describe in the same time. This 

 is a particular case of what is called the principle of virtual velocities. 



228. Let us suppose a very small motion given to the ma- 

 chine, and that the points of application of the two forces, describe 

 curves to which the directions of these forces are tangents. Let 

 u denote the velocity of the force />, or the space described in any 

 given time by />, and v the corresponding velocity of <?, or the 

 space described by q in the same time ; if the ratio of q to />, nec- 

 essary to an equilibrium, is required, we shall obtain it very near- 

 ly by the proportion 



q : p :: u : i>; 



and this will approach so much the more nearly to the exact 

 ratio, according as the motion impressed upon ihe machine is less, 

 Trig, 16. so that by taking the limit of the ratio of u to i>, we shall have 

 exactly the ratio sought of q top. 



If the directions of the forces q,p, are not tangents to the 

 curves described by the points of application of these forces, we 

 take, instead of the spaces described by these points, the projec- 

 tions of these spaces upon the directions of the forces, and the 

 inverse ratio of these projections will be the ratio of the forces, 

 in the case of an equilibrium. We are at liberty to take for the 

 point of application of each force, any point we may choose in 

 its direction, provided we regard it as firmly attached to the 

 machine. 



229. We shall now apply this rule to a few examples in order 

 to show its truth and utility. 



In the lever, I take for the points of application of the forces 

 rig<124 >, q, the feet L, M, of the perpendiculars PL, FM, let fall from 

 the fulcrum F, upon the directions of the forces. When the lev- 

 er turns about the point F, the points L, M, will describe the 

 similar arcs LZ/, MM', to which the directions^) L, qM, of the 

 forces, are tangents. The lengths of these arcs are to each other 

 Geom. as their radii FL, FM ; so that we have the proportion, 



OQD 



LU : MM' :: FL : FM, 



that is, 



u : v :: FL : FM', 



