
MR CLERK MAXWELL ON THE THEORY OF ROLLING CURVES. 533 
similarly if the operation be repeated m times, the resulting curve is 
t 
f) n+m 
i Dr+rm a cos ™ 
n+m 

When n = 1, the curve is 
r=2acos é 
the equation to a circle, the pole being in the circumference. 
When n = 2, it is the equation to the cardioid. 
r=4a (cos 5) 
2 
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 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 
ee (cos)! 
B 
+ 
9 
As the operation of rolling a curve on itself is represented by changing 7 into 
n + 1 in the equation, so this operation may be represented by changing » into 

n+ 4. 
Similarly there may be many other fractional operations performed upon the 
curves comprehended under the equation 
r= a (cos —) 
n 
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, = 2a sec 0, 

