[Invert with respect to O. We may derive other theorems' by inverting
properties of a circular cubic, e.g. Ch. XIV, § 3, Ex. 10, 11.]

Ex. 3. The circles of curvature of a bicircular quartic at its four
intersections with any circle for which it is self- inverse have four- point
contact.
Ex. 4. If O is the point half-way between the real singular foci of
a bicircular quartic, show that :

(i) O is half-way between the unreal singular foci, and the sum of
the squares of the distances of O from the singular foci is zero.
(ii) The points of contact of the tangents from O lie on a conic.
(iii) O is equidistant from the middle points of AB and CD, where
A, B, C, D are the intersections of any line with the quartic.
[Use equation (i) of § 3, and write down the first polar of O.]

Ex. 5. The bisectors of the angle between any pair of 'bitangents of
a bicircular quartic are parallel and perpendicular to the line joining
the real singular foci.
[Equation (ii) of § 3 has no term in xyi\

Ex. 6. The four intersections of two pairs of bitangents lie on a circle
whose centre O is half-way between the real singular foci.
The sum of the squares of the radii of the circles obtained by taking
two pairs of bitangents and also the other two pairs is the same for each
way of dividing into pairs.
X 2



308 BICIRCULAR QUARTICS XVIII 3

Ex. 7. The eight points of contact of any two of the conies (ii) of § 3
lie on a conic.
The eight points of contact of any two pairs of bitangents lie on a conic.
