Motion along curved Surfaces. 213 



or, 



vdv = gdx. 

 The integral of this equation is, . Cal - 



whence, 



v 2 = 2 O 2gor. 



In order to determine the constant C, let us suppose that the 

 point Ji from which the body begins to fall, is elevated above a 

 horizontal line passing through B by a quantity 13Z = h. It is 

 necessary, therefore, when v is zero, that x should be equal to h ; 

 accordingly we have 



= 2C 

 and consequently 



whence, by substitution, 



2 go? = 2g (/i a;), 

 = 2g X ZP. 



Now if a heavy body fall through the space ZP, the square of 

 the velocity which it will have upon arriving at P, will be 



2, X ZP. - 



Therefore, when a body descends along any curved line, it has at 

 any point whatever, the velocity which it would have acquired by fall- 

 ing freely through a space of the same perpendicular elevation. 



Thus the velocity which a body successively acquires by its 

 gravity in descending along the concavity of a curved line, is 

 altogether independent of the nature of this curve. 



340. Hence, if the body, after having arrived at the lowest 

 point 13 (the tangent to which I suppose to be horizontal,) meets 

 the concavity of the same or of any other curve, touching the 

 first in B, it will rise upon this last to a height equal to that 

 from which it descended. 



