134 
Proceedings of the Royal Society of Edinburgh. [Sess. 
Thus the series of pedals may be continued in both directions. They 
will be all changed if the position of O is altered. 
Cor. 3. The tangents at P, P p . . . make the same angle with the 
corresponding radii vectores OP, OP p . . . 
Cor. 4. If C passes through the pole O so do all the pedals. 
Cor. 5. If OP is normal to C at P all the pedals pass through P and 
have there a common tangent. 
Cor. 6. Since OP x L OP, .*. when C is a finite closed curve so are all its 
(positive) pedals. 
Cor. 7. If C has a parabolic branch so have the pedals. This does not 
happen for a hyperbolic branch of the curve. 
Cor. 8. When the pedal for 0 is known the pedal for O' may be 
found thus : 
Draw P X S l r OP p and O'S H 1 OP p then S is on the pedal of O'. 
§ 49. Prop. X. 
The Pedal of the Circle. 
Properties of the Pedal. 
{The Limacon of Pascal, and the Cardioid.) 
Let ATB be the given circle of centre C and radius r. Let OC = d, 
and describe the circle with centre O and radius OC. TP is the tangent 
at T and OP l r TP cuts the second circle in Q. Q q is ± r OC, and 
OF = OF' = r. 
Then 
OP = F q. 
For 
0 P = CT - OC cos 0 = r - d cos 0 = FO - OQ cos 0 = Fq. 
Equation to Pedal. 
Let 0 be the origin, C the point (d, 0). 
The equation to PT is of the form 
(x — d) cos $ + y sin c/> - r = 0 . 
y cos cj> — x sin <f> = 0 
and OP is given by 
• ( 1 ) 
• ( 2 ) 
