388 EPICYCLOID. HYPOCYCLOID. 



It may be found by a method similar to the above that for 

 the hypocycloid 



x = (a 5) cos 6 + b cos v - 0, 



y = (a b] sin b sin 7 6. 



361. The radius of the rolling circle may be greater or 

 less than the radius of the fixed circle both in the epicy- 

 cloid and in the hypocycloid ; it is however easy to infer 

 from the figure, that a hypocycloid in which the radius of the 

 rolling circle is greater than the radius of the fixed circle may 

 be counted as an epicycloid. This can also be shewn from 

 the equations. For in the equations to the hypocycloid put 



r 6=<f>', then those equations may be written 



, , , , ,7 N a + b a 

 x = (a + b a) cos d> (b a) cos -,- <f>, 



a 



/i ^ . a+b a 

 (b a) sin . <f) ; 



o ~ ct 



these are the equations to an epicycloid in which the radius 

 of the fixed circle is a, and the radius of the rolling circle 

 is b a. 



Similarly we may shew that a hypocycloid in which the 

 radius of the rolling circle is greater than half the radius of 

 the fixed circle may be counted as a hypocycloid in which the 

 radius of the rolling circle is less than half the radius of the 

 fixed circle. Thus we can obtain all epicycloids and hypo- 

 cycloids if in addition to epicycloids we take hypocj'cloids in 

 which the radius of the rolling circle is less than half the 

 radius of the fixed circle. 



362. If a and b are in the proportion of two whole num- 

 bers we may eliminate between the two equations which 

 determine an epicycloid or a hypocycloid, and thus obtain the 

 equation to the curve in an algebraical form. For example, 

 suppose in the hypocycloid that a = 4& ; then 



