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

22

THEORY OF CURVES.

if A..., F... denote the second differential coefficients of F (I, r, g),
we must have
Hf+ Bg + Fh =.....................(26),
Gf + Fg + Ch = O0
which requires that H (:, v, g) = 0. This shows that if the first
polar of a curve has a double point at B, then B must lie on the
Hessian; and therefore by ~ 36, the polar conic of the double
point B must break up into two intersecting straight lines. The
polar conic of B is
Aa2 + B32 - + Cy2 + 2F/y + 2Gyca + 2Hca3 = 0,
and the double point, which is the point of intersection of the two
straight lines constituting the conic, is determined by the equations
Aa + H3 +Gy = 0 &c. &c.,
which by (26) are obviously satisfied by (f g, h).
39. Equations (26) give relations between the coordinates of
the points A and B; and if we eliminate (I,, I) we shall obtain
the locus of A, which is called the Steinerian after the German
mathematician Steiner. The Steinerian is the locus of the points
of intersection of each pair of straight lines which is the polar
conic of points on the Hessian.
40. Every curve of the nth degree has n real or imaginary
asymptotes.
Since an asymptote touches the curve at infinity, it follows
that the asymptotes are the tangents at the points where the line
at infinity cuts the curve, and there are consequently n asymptotes.
A more analytical proof is furnished by the method for finding
asymptotes explained in books on the Differential Calculus. This
method consists in substituting ux + /3 for y in the Cartesian
equation of the curve, and equating the coefficients of xn and xnto zero, which furnishes two equations for determining Mu and 3.
Since the equation for /L is in general of the nth degree, i real or
imaginary values of,L exist.