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 inter 
sections, 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 
oi 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.