EPICYCLOID. HYPOCYCLOID. 



The arc NO arc MN and therefore = AN, 

 thearcO<7=JV7= CB. 



387 



therefore 



Hence is a point in a cycloid generated by rolling a circle 

 of radius a along BC. Hence the evolute of the cycloid 

 AEA is composed of the two semi- cycloids AB and A'B. 



360. The epicycloid is the curve traced out by a point in 

 the perimeter of a circle which rolls on the outside of a fixed 

 circle. 



Let and C be the centres of the fixed and the rolling 

 circles respectively, B the point of contact, P the tracing 

 point, A its initial position. Take OA as the axis of x ; 

 draw CN, PM, perpendicular to the axis of x. Let 



OB = a, BC=b, AOB = B, BCP=<I>. 

 Then x=ON+NM 



= (a + V) cos 9 + b sin (< - \ ir + 6) 

 = (a + b) cos 6 - b cos (0 + 0). 



But the arc AB = the arc BP, by the mode of generation, 

 that is, ad = b(f>, therefore 



x = (a + b) cos b cos y d. 



Similarly y = (a + 6) sin 6 b sin -. 0. 



The hypocycloid is the curve traced out by a point in the 

 perimeter of a circle which rolls on the inside of a fixed circle. 



CC2 



