38C 



EVOLUTE OF THE CYCLOID. 



Since the arc AM= afl, it follows that 



MP=zrcAM. 

 From (1) we have 



_, a x .. 

 y = a cos -f v (2o 



dy //2ax\ 



therefore -/- = * / . 



ax V \ x / 



If s denote the arc AP, we have 



(2), 





therefore 5 = *J(8ax], 



as will appear from the Integral Calculus. 



The normal to the curve at P is parallel to MD, as we 

 may see from Art. 357 or from an independent investigation. 

 By the property of the circle it follows that 



r\ 



MD = 2a cos - . 







If we investigate the value of the radius of curvature at P 

 we shall find it to be twice MD, that is, 



a 

 4a cos - , or 2 V(4a 8 2ax). 



Ji 



359. The evolute of the cycloid is an equal cycloid. 



For it appears by Art. 358 that the radius of curvature at 

 a point M of a cycloid is twice MN. Hence if we produce 

 MN to 0, making NO = MN, the point is the centre of 

 curvature corresponding to the point M, Draw EIB and 

 make IB = 2a ; draw B G parallel to ED ; the circle described 

 on NC as a diameter will pass through 0. 



