CYCLOID. 

 357. From the last equation we have 



385 



dy 

 Hence the equation to the tangent at M is 



2a y\ , , . 



- 



and the equation to the normal at M is 



y' -y = - /(-} (x' - X). 



V \ l 2a,-yj v 



If in the last equation we put y = 0, we have 

 x x = V{y (2 y}} = a sin < = PB. 



Hence MB is the direction of the normal at M, and therefore 

 M C is the direction of the tangent at M. 



If in the equations of Art. 356 we put < = TT, we have y = 2a 

 and x = air as the co-ordinates of the vertex E. Hence 



PD = air a<f> + a sin <j> 



Also the distance of M from a straight line through E parallel 

 to Ax is 2a a (t cos 0) or a (1 cos 6). 



358. If we take the vertex as the origin, and the tangent 

 at that point as the axis of y, we have by the last Article 



Describe a semicircle on AD as diameter : let PN meet 

 this circle at M, and join J/ with the centre 0; then 



AN=a(l-cosAOM); 

 therefore AOM= 0. 



T. D. c. 



cc 



