206 Dynamics 



326. Since the force GA passes through the centre of gravity 

 G of the body m, it must distribute itself equally to all parts of 

 this body. Therefore, so long as friction is supposed to have 

 no influence, the body can have no motion except that of sliding 

 along the plane, that is, it can have no tendency to roll, whatev- 

 er may be its figure, provided the perpendicular GB meets the 

 plane in a point that belongs at the same time to the surface of 

 the body. This would not be the case, however, as we have 



196> seen, if the perpendicular did not meet the base of the body, or 

 the surface by which it rests upon the plane. J"he influence of 

 friction, moreover, tends to produce a rolling motion. 



327. Since the body m must describe GA in the same time 

 in which it would describe GB by the free action of gravity, if 

 we conceive that at the end of the first instant, gravity acts anew ; 

 as it communicates in equal instants equal degrees of velocity, 

 by supposing for the second degree of velocity communicated in 

 a vertical direction, a decomposition similar to that above made 

 for the first instant, it is evident that the second parallelogram 

 will be equal in all respects to the first. We accordingly in- 

 fer, in like manner, that the force perpendicular to the plane 

 will be destroyed, and the force parallel to the plane, and equal 

 to GA, will be added to GA. By reasoning in the same manner 

 for the following instants, Ave should conclude that the velocity 

 along the inclined plane is accelerated by equal degrees ; in other 

 words, that the motion of heavy bodies down an inclined plane is a mo- 

 tion uniformly accelerated. Hence all that has been said upon the 

 subject of motion uniformly accelerated, is strictly applicable to 

 the motion that takes place down inclined planes. Consequently 

 in this latter case, as well as in the former, the velocities are as 

 the times, the spaces described are as the squares df the times, 



264, &c. or as the squares of the acquired velocities, &c. 



328. Therefore, in order to determine the motion that takes 

 place upon a plane of a known inclination, we have only to find 

 the ratio of the accelerating force to gravity, that is, the ratio of 

 GA to GB. Now GA, GB, being parallel respectively to DE, 

 DF, the angle AGB is equal to EDF, and the angle Jl being a 

 right angle as well as the angle F, the two triangles AGB, EDF, 

 are similar ; whence, 



