FORM OF TANGENTIAL EQUATION. 
403 
its reciprocal with respect to a circle whose radius is Jc is another equiangular spiral 
whose equation in polar coordinates is 
Cor. The positive and negative pedals of equiangular spirals are also equiangular 
spirals, and so is the inverse. So that every geometrical transformation of this curve is 
another curve of the same species. 
Section III. — Reciprocal Curves. 
62. We have seen that the polar equation of the first positive pedal of the curve 
e=/t<p) sin <p ; 
IS 
and the reciprocal of a curve being the inverse of its first positive pedal, then the polar 
equation of the reciprocal of v=f(q>) is 
k l 
i 
— /(<p)sin<p 
(178) 
Thus the reciprocal of the parabola is 
(179) 
or, in Cartesian coordinates, 
(180) 
which is another parabola, as it ought, since the centre of reciprocation is a point on 
the curve. 
63. Since the value of § derived from art. 178 is 
S At) sin 
we infer, from art. 55, that the equation of the first negative pedal of the curve is 
At) sin V 
Hence we have the following theorem : — If v=f(<p) be the tangential equation of a curve 
the reciprocal of its first positive pedal or the first negative pedal of its reciprocal is 
V= At) sin 2 <p ( 181 ) 
64. If the intrinsic equation be given, say s=F(<p), then we have, from equation (167), 
the polar equation of its reciprocal, 
( 182 ) 
