MR CLERK MAXWELL ON THE THEORY OF ROLLING CURVES. 533 



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



cos "'" \ 

 n + ??i / 



When H = 1, the curve is 



r — 2 a cos S 



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

 When n = 2, it is the equation to the cardioid. 



/• = 4 o I cos — 1 



In order to obtain the cardioid fi-om the circle, we roU the circle upon itself, 

 and thus obtain it by one operation ; but there is an operation which, being per- 

 formed on a circle, and again on the resulting curve, will produce a cardioid, and 

 the intermediate curve between the circle and cardioid is 



r = 2 a I cos — 1 ' 

 \ 3 J 



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

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



n + 



n + ^. 



Similarly there may be many other fractional operations performed upon the 

 curves comprehended under the equation 



»• = 2" a ( cos — \ 



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

 curve, by making n — — 1. 



We may likewise prove by the same method as before, that the result of per- 

 forming this inverse operation an infinite number of times is the logarithmic 

 spiral. 



As an example of the inverse method, let the traced line be straight, let its 

 equation be 



r^ =: 2 a sec 6^ 

 P-\ _ Pd _ 2a _ 2a 



then 



r r 



-1 '0 r„ 2/>_j 



p--i = a r_^ 



