138 
Proceedings of the Royal Society of Edinburgh. [Sess. 
Cor. 3. If the generating curves are ellipses, the focus of the moving 
-ellipse generates a circle, and any other point a bicircular quartic. 
Similar conclusions hold for the hyperbola. 
or 
§ 52. The curves whose radial equation can be represented in the form 
p = Ar n+1 
p/r = (r/a) n . 
These curves have the property that their pedals for the same pole 
have a similar radial equation. 
Let p 1 and r 2 be the elements of the first positive pedal corresponding 
to p and r of the given curve. 
Then 
pJ r i =p/ r 
and 
But 
and finally 
i\ = p; 
V = r i, r = r 1 2 /p 1 . 
p/r = ( r/a ) n , 
••• Pi/ r iT-Wl a Pi)"l 
/',/>■] - ()'i/“)" +I • 
Similarly the second positive pedal is given by 
Pj r 2 = ( r J a ) n,(2n+1) 
and the mth pedal by 
Pmb'm = ( r ml a ) mn+1 • 
Similarly, the mth negative pedal is given by 
Pm = (pmid )~ mn+1 
Particular examples of p — r equations are : — 
(I.) Circle of radius ,a, the pole being on the circumference, 
p/r=-rl'2a (n = 1). 
(II.) The straight line at a distance a from the pole, 
p/r =■ a/ v (n = — 1). 
( 2 ) 
(3) 
(4) 
(5) 
( 6 ) 
(III.) The Parabola (first negative pedal of the straight line), 
p/r = (r/a)~% 
(IV.) The Rectangular Hyperbola, 
