384- SPIRAL OF ARCHIMEDES. CYCLOID. 



355. The Spiral of Archimedes. 



356. The Cycloid. 



The cycloid is traced out by a fixed point in the circum- 

 ference of a circle as the circle rolls along a straight line. 



Let Ax be the straight line along which the circle rolls ; 



M the fixed point in the circumference of the circle 

 BMC which traces out the cycloid ; 



A the point in the straight line Ax with which M. 

 was originally in contact ; 



the centre of the circle : 

 AP = x, MP = y, MOB = (j), OB = a. 



The arc MB = a<f>, and by the nature of the curve it is 

 equal to AB ; 



therefore x = a,(f> PB = a< a sin <>, 



y = a a cos <p. 

 If we eliminate <f> we have 



_, a it , ._ ,. 



x = a cos \(2ay - y ) 



