Line of swiftest Descent. 227 



then, by integrating, we have 

 dy ds 



or 



u O 



But since the velocity u is equal to that which the body would 

 acquire in falling from the height AP, we have 



u* = 2gx. 277 ' 



Therefore, 



Cdy = ds V %g x i 

 and 



C 2 dy 2 = 2gxds 2 = 2gx(dx* + dy 2 )-, 



whence we deduce 



To determine the constant C, it will be observed that when 

 = C, we have dy ds, answering to the lowest point R, of 

 the curve, where d a?=0, and xh ; therefore if we call v the velo- 

 city which the body will have at the point where v %~gx = C, the 

 equation Cdy = uds becomes C d s = v d s, which gives C = u. 

 And if we call h the corresponding height ^fC, we shall have 



V 2 - 



hence, 



C 2 = 2g/i. 

 Therefore, 



, dx/'Zx dx 



V ggz V h x ' 



is the equation of the curve. But the better to understand this 

 curve, let us give another form to the equation. 



Imagine the vertical line RD drawn through the point /?, 

 where dy = d s ; and having produced PM to O, call AD, a ; 

 OR,x f , and OM,tf. Thenx = h tf,y = a y,da? = 

 ^ t/ = d\f ; substituting these values, we have 



a?' d a? 7 



x' 



A a;' 





