An elementary treatise on cubic and quartic curves, by A. B. Basset.

PEDAL CURVES. INVERSION.

39

k2x/r2, k2y/r2 for x and y; hence u,. becomes k22U,,/r2n, and the
equation of the inverse is
k222tu + k2n-2r2t_ +... k2r.-22ut + r21ut0 = 0......(19).
The degree of the inverse of a curve of the nth degree is in
general 2n; but if the origin be a multiple point of order k, the
degree of the inverse will be 2n-k. The degree will be still
further reduced if un, un-_ &c. contain some power of r as a factor.
Since the degree of the reciprocal polar is equal to the class of
the curve, the degree of the pedal can be found by inversion.
65. To find the Cartesian equation of the pedal of a curve.
Let 0> (I, 7) = 0 be the tangential equation of a curve; let any
tangent cut the axes of x and y in A and B; also let (x, y) be the
coordinates of Y, the foot of the perpendicular from 0 on to the
tangent AB. Then if
AOY= 0,
0 Y= OA cos 0 = OB sin 0,
whence
x- n2Jl + 12,
and
x y
- =.2 + y2 2 y2'
Hence the Cartesian equation of the pedal is
x {2+y2 2+ y2} =0
66. To find the tangential equation of the first negative pedal.
If F(x, y)= 0 be the Cartesian equation of the curve, it
follows from the preceding formulae, that the required tangential
equation is
F{2 2 ' t2+D2}=By means of the preceding results, it may be shown that the
Cartesian equation of the first positive pedal of the curve
(x/al)' + (y/b)h = 1
is
n n n
(x2 + y2)1 - = (a.)nY-+ (by)-1,
and that the tangential equation of its first negative pedal is
( y2 + 772)n = (/a)'2, + (v/b)n.