146 OF DEFLECTIVE FORCES. 



the effects of accelerating forces, without the inconveni- 

 ence of too great a velocity; although, if the weights 

 employed roll down the plane, some force will be lost in 

 the production of rotatory motion ; and if they slide, they 

 will be retarded by friction. 



284. Theorem. "256." When bodies 

 descend on any inclined planes of equal height, 

 their times of descent are as the lengths of the 

 planes, and the final velocities are equal. 



2r 1 



Since f:= V (—) (233), and here a=:^, tzz s/(2x^) = 



s/2x; and the times vary as the spaces: but the times 

 being greater in the same proportion as the forces are less, 

 the velocities acquired are equal (230). 



Scholium. Thus a body will acquire a velocity of 32 

 feet in a second, after having descended 16 feet, either in 

 a vertical line or in an oblique direction ; but the time oi^ 

 descent will be as much greater than a second, as the 

 oblique length of the path is greater than 16 feet : and if 

 we suffer three balls to descend together along three 

 grooves of the same height, but of the lengths of 1, 2, 

 and 3 feet respectively, we may estimate by the ear the 

 equaUty of the intervals at which they reach the bottom. 



285. Theorem. "257." The times of 

 falling through all chords drawn to the lowest 

 point of a circle are equal. 



J^L^^ The accelerating force in any chord A B 



is to that of gravity as A C to A B, or as 

 A B to A D (121), therefore the forces 

 being as the distances, the times are equal ; 



