210 Dynamic's. 



DFxt' 

 ~DE~' 



substituting for t this value in the above equation, we shall have, 



V = Urn 



Therefore, if several bodies descend along planes differently inclined, 

 but of the same height, they will have the same velocity upon arriving 

 at the same horizontal line. 



Of Motion along curved Surfaces. 



335. If a body without gravity and without elasticity, des- 

 cribes, in virtue of a primitive impulse, the successive sides JlB, 



Fjg.160. #Q - c ^ O f ar) .y. p0iyg 0nj U p 0n meeting each side it will lose a 

 part of its velocity, which may be determined in the following 

 manner. 



Let us suppose that the body moves from A toward B, and 

 that when it is at B, its velocity is such as in a determinate time, 

 one second for example, would cause it to describe, if it were 

 free, the line BF in JIB produced. Having erected upon BC 

 from the point B, the perpendicular BE, we imagine the rectan- 

 gular parallelogram BDFE, of which BF is the diagonal, and 

 the sides of which are in the direction of BC and BE. Instead 

 of the velocity BF, we may suppose that the body has at the 

 same time the two velocities BD, BE ; and as the side BC pre- 

 vents its obeying the velocity BE, it is manifest that its velocity 

 is reduced to BD. 



If from the point B, as a centre, and with a radius BF, we 

 describe the arc FI, DI, which is the difference between BF and 

 BD, will accordingly be the velocity lost. Now DI is the versed 

 sine of the arc FI, or of the angle FBC, made by the two con- 

 tiguous sides AB, BC. Therefore so long as these two sides 

 make a finite angle, the body will lose a finite part of its veloc- 

 ity upon meeting each of the sides. 



336. But if the angle formed by the two sides is infinitely 

 small, the velocity lost will not only not be a finite quantity, but 



