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

24

THEORY OF CURVES.

whence if A be a point of inflexion, the polar conic is
{(n - 1) a + v, = 0..................(29),
from which it appears that the polar conic of a point of inflexion
breaks up into two straight lines, one of which is the tangent
u, = 0, whilst the other is the line (n - 1) a + vi = 0. Hence every
point of inflexion is a point on the Hessian. Also since the
degree of the Hessian is 3(n - 2), the number of points of inflexion
cannot exceed 3n (n - 2).
If in (27) all the coefficients up to and including Uk-_ are zero,
the vertex A is a multiple point of order k; and the equation
uk= 0 determines the k tangents to the curve at A.
43. If a curve has a multiple point of order k, that point will
be a multiple point of order k - 1 on the first polar, of order k - 2
on the second, and so on.
Let A be the multiple point and B the pole. Then the
equation of the curve is of the form
uka-k + Uk+ -- +.., =............(30),
and the first polar of B is
du-kn + duk+ -k- dun
_ a k~ an-k- =,
d/3 d3 d=0
and since duk/d/3 is a binary quantic of degree k-1, it follows
that A is a multiple point of order k - 1 on the first polar.
44. If two tangents at a multiple point coincide, the coincident
tangent touches the first polar of every point.
The equation uk=O gives the k tangents at the multiple
point A; but if two of them coincide, we must have
Uk = (,af v+) vk-2.
Now the coefficient of an-k in the first polar of B is
(/u3 + vy) {2/vk_2 + (,3 + vAy) dvk_,/d/},
which equated to zero gives the tangents at A to the first polar;
hence the line,8/ + vy = 0 touches both curves.
Putting k = 2, it follows that the tangent at a cusp touches
the first polar of every point.