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

FOCI.

91

determines the two remaining single foci, which are the inverse
points of the foci of the conic.
In the case of the cissoid a = 0, and the equation becomes
a (a - 4b) = 0,
which shows that the cusp is a triple focus composed of three
single foci, whilst the other single focus is the inverse of the focus
of the parabola. The double focus is determined by the equation
x =-b.
147. If 0 be the node of any nodal circular cubic, S and H
the two single foci, and P any point on the curve,
1. SP + m. HP = n. OP,
where 1, m, n are constants. Also if a central conic be inverted
with respect to its vertex 0, and A be the vertex of the cubic,
SP HP OP
OS OH OA '
Let S', H' be the foci of the conic, 2A its major axis, P' any
point on the conic; then, if unaccented letters denote the inverse
points,
SP S'P' S'P'. OS
OP OS' 2 '
HP H'P'. OH
OP - k
SP HP OP
whence SP O + OH = kP (S'P' + H'P')
- 2A. OP/12,
which proves the first part. But when 0 is the vertex of the
conic, k2/2A = a = OA, which proves the second part.
148. The following propositions may be proved by inversion
for any nodal circular cubic.
(i) The circles passing through OSP and OHP cut the curve
at equal angles.
(ii) If any circle passing through 0 and a focus cut the cubic
in P and Q, the tangent circles at P and Q which pass through 0
intersect on a fixed circle.