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

MULTIPLE POINTS.

25

45. A multiple point of order k on a curve is a multiple point
of order 3k- 4 on the Hessian.
The equation of a curve having a multiple point at A is given
by (30), and if A = d2F/da2..., F= d2F/d3d7y..., the equation of
the Hessian is
ABC + 2FGH- AF2- BG2 - CH2= 0.........(31),
which is of degree 3n- 6. Now the degrees of a, /, 7y in the
different terms are shown in the following table:
A B C F G H
a n-k-2, n-k, n-k, n-k, n-k-1, n-k-1,
/3 k, k-2, k, k, k, k-1
7 k, k, k-2, k-2, k-1, k
From this table it appears that the highest power of a is of
degree 3n - 3k - 2, and that its coefficient is a binary quantic in,/ and y of degree 3k -4. Hence A is a multiple point on the
Hessian of order 3k - 4.
46. Every tangent at a multiple point on a curve is a tangent
to the Hessian at that point.
Let the line / = 0 coincide with any tangent through A to the
curve; then Uk must contain 8/ as a factor and must therefore be
equal to /vki. But on referring to the table we see that the
highest powers of a in A, C and G must contain /3 as a factor,
and since every term of the Hessian must contain A, C or G, the
coefficient of the highest power of a in the Hessian contains /
as a factor and therefore this line is the tangent at the point A
to the Hessian.
Putting = 2, it follows that every double point on a curve is
a double point on the Hessian, and that the tangents at the
double point are common to the curve and its Hessian.
Singularities at Infinity.
47. In ~ 41 we investigated the conditions that a curve
should have a double point or a point of inflexion at a finite
distance from certain lines of reference; but it frequently happens
that a curve has singularities at infinity, and we shall now explain
a method by which such singularities may be determined.