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

POLAR CURVES.

19

the curve is equal to m; and the preceding results show that if 0
is a node the number of tangents is equal to m- 4; if 0 is a cusp
or a point of inflexion, the number is mn-3; whilst if 0 is an
ordinary point on the curve, the number is m - 2.
29. The equation
APF(a,, /3,y)=0.............. (20)
is called the pth polar of the curve with respect to (f, g, h), and is
a curve of degree n-p. Also by ~ 23, the pth polar may be
written in the form
n-F(f g, h)= 0..................... (21).
The (n - i)th polar is therefore a straight line, which is called
the polar line; whilst the (n-2)th polar is a conic, which is
called the polar conic. The equations of the polar line and polar
conic are
'F' = 0, and A'2F' = 0................. (22).
If one of the vertices, say A, of the triangle of reference be
taken as the pole, g = h = 0, and the pth polar assumes the simple
form
=...........................(23).
daP
By means of (19) of ~ 24, it can be shown that when a curve
is expressed in terms of Cartesian coordinates, the first polar of
(f g) is
dF dy
(0a, g+ gdd - +Un-i + 2ln2+... nu,=0. a. (24).
30. The locus of all points, whose polar lines pass through a
fixed point, is the first polar of that point.
Let (f, g, h) be the fixed point; (:, V7, ~) any other point.
The equation of the polar line of (:, v, r) is
dF dF dF
ad +1d + =0-;
but if this pass through the point (f, g, h)
dF dF dF
f 4+g +h dd =0,
which shows that the locus of (:, r, ') is the curve AF = 0, which
is the first polar of (f, g, h).
2-2