208 Dynamics. 



whence, 



DA : DB :: 



:: DE X t 2 : DF X t' 2 . 



But by similar triangles, 

 Gfjom. 



213. DA : DB :: DE : DF; 



consequently, 



DE : DF : : DE X t* : DF X *' 2 ; 

 therefore, 



/'2 _ j 2 or // = (f 



Fie.159. 331. Hence, if DG be a third plane described by a third 

 body setting out from D at the same time with the other two, by 

 drawing from the point A the perpendicular AC, A, B, C, are 

 the three points at which the three would arrive in the same 

 time. 



332. If upon DA as a diameter, we describe a semicircum- 

 ference, it will pass through the points C and B, since the angles 



Geom. at Q anc j are r ight angles. Consequently, the chords 1)C, 

 DB, and the vertical diameter DA, are all described in the same 

 time ; and as this does not depend upon the length or inclination 

 of the chords, we may draw the general conclusion, that the time 

 employed by a body in falling through any chord of a circle, drawn 

 from the extremity of a vertical diameter, is the same as that employ- 

 ed in falling through this vertical diameter. 



333. We have seen that g being the velocity communicated 

 to a free body in one second, is that given in the same 



time to a body that descends along DE. Let t, t', be the times 

 employed in describing DF and DE respectively; we shall 

 have 



whence, 



g -~^ X 



