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

90

SPECIAL CUBICS.

ones; also the inverse points of the foci of the conic, whose inverse
the curve is, are two of the single foci; and the node is a third
focus. On the other hand, cuspidal cubics are of the third class,
but in consequence of the cusp replacing the node, the curve has
the same number of double and single foci. It will be shown in
Chapter VIII. that when a circular cubic has a double point, the
latter is a double or a triple focus composed of the union of two or
three single foci, as the case may be, according as the double point
is a node or a cusp; but for the special class of circular cubics
considered in the present Chapter a direct proof may be given
as follows.
Transform the cubic x (x2 + y2) = ax2 + by2 into trilinear coordinates by taking an imaginary triangle of reference, one of
whose sides is the line at infinity, whilst the other two sides are
the lines joining the double point with the circular points. Then
we may write
/3=x+ty, ry=x-ty, I=1,
and the cubic becomes
23y (/3 + y) = {a (/3 + )2 - b (3 - y)2 I,
and therefore the tangents at the circular points (ry, ), (I, I) are
2y=I(a-b), 2/3=I(a-b),
or in Cartesian coordinates
2 (x - y)= a - b, 2 ( + y)= a - b,
which intersect at the point 2x = a - b, y = 0, which determines
the double focus.
To obtain the real single foci, we observe that symmetry shows
that they must lie on the axis of x; we must therefore find the
condition that the line x - a + y = 0 should touch the cubic,
where (a, 0) are the coordinates of any focus. The points of
intersection of this line with the cubic are determined by the
equation
x2 (a - b - 2a) + a (2b + ag- ba2 = 0,
and the line will be a tangent if
a2 (a2- 4ba + 4ab) = 0.
In the case of a nodal cubic, the factor a2 = 0 determines two
of the single foci, which shows that the node is a double focus
formed by the union of two single foci; whilst the other factor