20 THE THEORY OF ROLLING CURVES. 



n n 



Now 0,-0.= -cos- l ^= -cos' 1 cos -* = -', 



7", 



-0 

 "'- 



substituting this value of 0, in the expression for r,, 



\" +1 



similarly, if the operation be repeated m times, the resulting curve is 



/ 6 \ n+m 

 r m = 2 n+m a ( cos - 



When n=l, the curve is 



r = 2a cos 0, 



the equation to a circle, the pole being in the circumference. 

 When w = 2, it is the equation to the cardioid 



r = 4 (cos^) . 

 \ / 



In order to obtain the cardioid from the circle, we roll the circle upon 

 itself, and thus obtain it by one operation ; but there is an operation which, 

 being performed on a circle, and again on the resulting curve, will produce a 

 cardioid, and the intermediate curve between the circle and cardioid is 



? = 



As the operation of rolling a curve on itself is represented by changing n 

 into (n+1) in the equation, so this operation may be represented by changing n 

 into 



Similarly there may be many other fractional operations performed upon 

 the curves comprehended under the equation 



f)\ 



/ f)\ n 

 =2"a (cos - ) . 

 \ V 



We may also find the curve, which, being rolled ou itself, will produce a 

 given curve, by making n= 1. 



