Line of swiftest Descent. 229 



352. We have supposed the body to have no velocity on its 

 leaving the point A. But if it had already acquired a certain 

 velocity in a given direction, the origin of the curve would be 

 at some higher point. The equation Cdy = u d s, found above, 



gives -^j- = -~ ; whence the constant C must be such that the 



initial velocity being divided by it, the quotient will be equal to 

 the sine of the angle made by the direction of this initial velocity 

 with the vertical, a condition, which, with the other, that the 

 body must pass through A and J2, will determine the cycloid for 

 the case in question. 



353. Besides this property of being the curve of swiftest 

 descent in an unresisting medium, the cycloid is on several other 

 accounts quite remarkable. It has, for example, this singular 

 property, that whatever be the point X, from which a body be- 

 gins to descend along the concave part of the curve, it arrives 

 always at the lowest point R in the same time* This property 

 is thus proved. 



Calling t the time, and s the arc RM corresponding to any 

 point M) where the body is found at die end of the time , we 



have d t . Now designating by h' the height of X 



above the horizontal line OM, we have u = v 2 g (h! x') . More- 277. 

 over, it is easy to infer from the value of d y', found above, that 



hence 



dx' 



x 



' 



therefore, 



x /_. . 



2 J Vh'x' x'z 



whence, reasoning as above, and calling ^ the whole time em 

 ployed in falling from X to jR, we conclude that 



