it not be proved to be a _circle_? That is to say, he will be more
conservative than ever. He would like to return to a primitive form of
government. Farewell to his wild schemes and revolutionary measures!
Farewell to his disestablishments, abolitions, and suppressions! The
throne and government have new attractions in his eyes; loyalty, a new
feeling, asserts its benign influence; and if he could return to his
former position, his normal conduct would be straighter than ever, for
by sad experience he has learned the value of those things which he once
despised.

But we need not depend upon one proof alone. Exactly the same result may
be obtained from the well-known proposition which states that 'the angle
between the tangent from any external point and the focal distance is
equal to the angle between the other tangent and the focal distance.'

3. The same opinions are often held by individuals in quite different
walks and classes of life. Let these individuals be represented by
points on an ellipse. Join these, and we have a system of parallel
chords. Draw a straight line through the middle points of these chords,
and lo! it will always pass through the centre. This shows that the
central thought of all people is directed to the sovereign--that
_loyalty_ is inherent in the hearts of those who recognise elliptical
laws.

I will conclude this lecture with a few remarks on the nature and
properties of the _radical axis_. This name was first given, I believe,
by M. Gaultier, of Tours, and for a full account of its nature I refer
you to the _Journal de l'Ecole Polytechnique_, xvi., 1813. The radical
axis of two circles is the line perpendicular to the line joining the
centres, from any point of which the tangents to the circles are equal.
Let us suppose that one circle becomes a point, and that this point is
situated on the circumference of the first circle. What is the result?
The radical axis becomes the tangent to the circle. Hence we may
conclude that in a social system of monarchical government the radical
axis is perpendicular to the line attaching the individual with the
monarch. Therefore we may conclude that the radical axis indicates a
tendency of particles, or individuals, to fly off at a tangent, at right
angles to the connecting-link between the individual and the king. When
any motion takes place, this is evident, and this tendency is called
centrifugal force. Sad is it for the State when this force is ca