The collected mathematical papers of Arthur Cayley.

542

ON POLYZOMAL CURVES.

[414

The centres of the two circles respectively are the two foci of the conic, which foci
lie on the axis in question. Observe that in the general case there are at each of
the circular points at infinity two tangents, without any correspondence of the tangents
of the one pair singly to those of the other pair, and there are thus four intersections, the four foci of the conic; in the present case, where the curve is a pair
of circles, the two tangents to the same circle correspond to each other, and intersect
in the two foci on the axis in question. The other two foci, or antipoints of these,
are each of them the intersection of a tangent of the one circle by a tangent of
the other circle.
If the conic has with the circle a contact of the third order (this implies that
the circle is a circle of maximum or minimum curvature, at the extremity of an axis
of the conic), then the curve has at this point a tacnode, viz., it breaks up into two
circles touching each other and the conic at the point in question, and having their
centres at the two foci situate on that axis of the conic respectively.
176. If the conic is a parabola, then the curve is a circular cubic having the
four intersections of the parabola and circle for a set of concyclic foci, and having
the focus of the parabola for centre. The like particular cases arise, viz.,
If the circle touch the parabola, the curve has a node at the point of contact.
If the circle has, with the parabola, a contact of the second order, the curve has
a cusp at the point of contact.
If the centre of the circle is situate on the axis of the parabola, then the four
intersections are situate in pairs symmetrically in regard to this axis, and the curve
has this axis for an axis of symmetry.
If the circle has double contact with the parabola (which, of course, implies that
the centre lies on the axis), then the curve has a node at each of the points of
contact, viz., the curve breaks up into a line and circle intersecting at the two points
of contact, and the circle has its centre at the focus of the parabola.
If the circle has with the parabola a contact of the third order (this implies
that the circle is the circle of maximum curvature, touching the parabola at its
vertex), then the curve has a tacnode, viz., it breaks up into a line and circle touching
each other and the parabola at the vertex, that is, the line is the tangent to the
parabola at its vertex, and the circle is the circle having the focus of the parabola
for its centre, and passing through the vertex, or what is the same thing, having its
radius =- of the semi-latus rectum of the parabola.
177. If the conic be a circle, then the curve is a bicircular quartic such that its
four nodo-foci coincide together at the centre of the circle; viz., the curve is a
Cartesian having the centre of the conic for its cuspo-focus, that is, for the intersection
of the cuspidal tangents of the Cartesian. The intersections of the conic with the
other circle, or say with the orthotomic circle, are a pair of non-axial foci of the
Cartesian; viz., the antipoints of these are two of the axial foci. The third axial
focus is the centre of the orthotomic circle.