214 Dynamics. 



Indeed, let us suppose that the body is actually in B, or 

 that x = ; its velocity will be such, that we shall have 



v 2 = 2g/i, or u 2 = 2g/i, 



by calling this velocity u to distinguish it from the other. Let us 

 imagine that with this velocity it ascends along any curve BM' ; 

 we shall find by the same reasoning as that above pursued, that 

 its velocity in any point M', is determined by the equation, 



by calling v', the velocity in this case, and s* the arc BM', and 

 observing that v' diminishes according as /', Y, and x increase 



280. respectively. Consequently, putting for d t its value r , we 

 shall have 



d t)' = ^___ or i/ d v' = g d x 5 



and by integrating, 



T / 2 - 2C 2gx. 

 But, when x = 0, the velocity t/ is u ; accordingly, 



u 2 = 2 C 0, 

 and since 



we have 



Now when the body ceases to ascend, v' = which gives 



whence we deduce 



Therefore the point at which the body will have arrived in any 

 curve BA 1 , will be at the same height as the point A. 



