Motion down inclined Planes. 209 



which gives, by multiplying the extremes and means and reduc- 

 ing, 



2 



DP 



X i'* = DE x * 3 , 



or, 



DF x V* = DE x t 2 , 

 or, taking the square root of each member, 



OF x t 1 = DE x f ; 

 in other words, 



t : t' :: DF : DE. 



In like manner, if f" represent the time employed in describing 

 DG ; we shall have 



t : t" :: DF : DG; 

 whence, 



*' : t" : : DE : DG ; 



that is, the times employed in describing different planes of the same 

 height, are to each other as the lengths of these planes. 



334. The velocity of the body which descends along DF, is 

 g t at the expiration of the time t. For a similar reason, the 2 i67. 



velocity of the body that descends along DE, is s * x t' at 



the expiration of the time t'. Accordingly, if we call w, v, the 

 velocities acquired by the two bodies respectively upon arriv- 

 ing at the points F, E, we shall have, 



whence, 



-ogt = ug X -=-& x *' 



But, as we have just seen, 



t : t 1 : : DF : DE, 

 which gives 



Mech. 27 



