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

58

CUBIC CURVES.

and therefore consists of two straight lines, one of which t, = 0 is
the tangent at the point of inflexion A, whilst the other line
2a + v1 = 0 passes through the points of contact of the tangents
from A. The latter line is called the Harmonic Polar of the
point of inflexion, and is a line of considerable importance in the
theory of cubic curves.
We shall now prove a more general theorem, of which the
preceding proposition is a particular case.
93. If a straight line intersect a cubic in three points D, E, F;
the three points D', E', F' in which the tangents at D, E, F intersect
the cubic lie on a straight line.
We shall first prove that every cubic can be expressed in the
form
uvw + k't'vi'w = 0.........(.......... (6),
where n, v, w and iu', v', w' are linear functions of (a, 3, y) and
therefore represent three straight lines.
The general equation of a cubic which passes through the
vertices of the triangle of reference is
a2it + Oaz +,3y (m/3 + ny)= O=
Add and subtract lat3y and the equation becomes
c (aut + It2. - l3y) + /3y (la + mf + 1ny) = 0();
the second term is the product of three straight lines, whilst the
first term is the product of a conic and a straight line. Now I
may have any value we please; if therefore we determine I so
that the discriminant of the conic vanishes, the first term will
also be the product of three straight lines.
Equation (6) accordingly represents a cubic passing through
the nine points of intersection of (u, v, w) and (u', v', w').
If u'=v', (6) becomes
uvw + ku/'tw' = 0....................(7),
which is the equation of a cubic which touches the straight lines
u, v, w at the points where t' intersects them; also the form
of (7) shows that the three points in which tu, v, w intersect the
cubic lie on the line wi'=0.
If D, E, F and D', E', F' be the points in which the lines u'
and w' respectively intersect the cubic, the points D', E', F' are