sects the line of centres; these
points are named the "limiting points." In the case of a coaxal system
having real points of intersection the limiting points are imaginary.
Analytically, the Cartesian equation to a coaxal system can be written
in the form x squared + y squared + 2ax +- k squared = 0, where a varies from member to
member, while k is a constant. The radical axis is x = 0, and it may be
shown that the length of the tangent from a point (0, h) is h squared +- k squared,
i.e. it is independent of a, and therefore of any particular member of
the system. The circles intersect in real or imaginary points according
to the lower or upper sign of k squared, and the limiting points are real for
the upper sign and imaginary for the lower sign. The fundamental
properties of coaxal systems may be summarized:--

1. The centres of circles forming a coaxal system are collinear;

2. A coaxal system having real points of intersection has imaginary
limiting points;

3. A coaxal system having imaginary points of intersection has real
limiting points;

4. Every circle through the limiting points cuts all circles of the
system orthogonally;

5. The limiting points are inverse points for every circle of the
system.

The theory of centres of similitude and coaxal circles affords elegant
demonstrations of the famous problem: To describe a circle to touch
three given circles. This problem, also termed the "Apollonian problem,"
was demonstrated with the aid of conic sections by Apollonius in his
book on _Contacts_ or _Tangencies_; geometrical solutions involving the
conic sections were also given by Adrianus Romanus, Vieta, Newton and
others. The earliest analytical solution appears to have been given by
the princess Elizabeth, a pupil of Descartes and daughter of Frederick
V. John Casey, professor of mathematics at the Catholic university of
Dublin, has given elementary demonstrations founded on the theory of
similitude and coaxal circles which are reproduced in his _Sequel to
Euclid_; an analytical solution by Gergonne is given in Salmon's _Conic
Sections_. Here we may notice that there are eight circles which solve
the problem.

_Mensuration of the Circle._

All exact relations pertaining to the mensuration of the circle involve
the ratio of the circumference to the diameter. This ratio, invariably
denoted by [pi], is constant for all circles, but it does not admit of
exact arithmetical exp