260 ON THE CONICS WHICH PASS THROUGH [342
Whereas, if the given line cuts two sides and a side produced, we have more
simply as in fig. 2.
Fig. 2.
But to gain a more precise knowledge, it is proper to consider the curve which
is the locus of the centres of the conics of the system.
Such locus which, as will presently be seen, is a curve of the fourth order, must,
it is clear, pass through the points of intersection (L 1} L,, L 3 in figs. 1 and 2 respectively
and p, q, r in fig. 3 presently referred to) of the sides with the given line; and it
is not difficult to show geometrically that it touches, at these points, the sides of the
triangle. It may be shown also that the curve has three nodes (double points), viz.
the middle point of each side of the triangle is a node of the curve. In fact if upon
any side as base we apply an equal and opposite triangle so as to form with the
given triangle a parallelogram, then any conic through the four vertices of the
parallelogram will have for its centre the central point of the parallelogram; that
is, the middle point of the side in question. But we may through the four vertices
describe two conics, each of them touching the given line; that is the middle point
of the side is the centre of two different conics of the system, and it is therefore a
node upon the curve of centres. And moreover the node will be a crunode or an
acnode (i.e. a double point with two real branches, or else a conjugate or isolated
point) according as the conics are real or imaginary: and it is easy to see that
if the given line does not cut the parallelogram, or if it cuts two opposite sides,
the conics will be both real; but if it cuts two adjacent sides the conics will be both
imaginary; that is, in the former case we have a crunode, and in the latter an
acnode. Through each node may be drawn two tangents to the curve; and it is a
known property of curves of the fourth order that the six points of contact lie on
a conic; one of the tangents through the node is however the side whereon the node
lies, and the points of contact of the three sides lie on a line, viz. the given line:
hence the last mentioned conic is composed of the given line, and another line; that
is, the three points of contact of the other tangents through the three nodes lie on
this other line.
It is proper to add that the points at infinity of the curve of centres are the
centres of the four parabolas; that is, there will be four infinite branches, if the
parabolas are real, viz. if the given line cuts the three sides produced; but no infinite
branch if the parabolas are imaginary, viz. if the given line cut two sides and a side
produced.
