An elementary treatise on cubic and quartic curves, by A. B. Basset.

246

THEORY OF PROJECTION.

turned through an angle I r becomes 4a2 (X2 + y2) = (x2 - y2)2. By
~ 188 this curve has a complex biflecnode at the origin and a pair
of real ones at infinity which lie on the lines y = + x, and if in (3)
of ~ 355 we put 1 = o, m = 1, the resulting formulae project the
two real biflecnodes into the circular points, and the curve projects
into the lemniscate 4a2 (y2- x2) (x2 + y2)2. Applying this projection to the theorem at the end of ~ 339, the circle becomes a
rectangular hyperbola, whence:If a pair of real tangents be drawn to a lemnwiscate from any
point on the curve, the envelope of the chord of contact is a rectangular hyperbola whose centre is the real node of the lemniscate.
In the general case of any quartic with three biflecnodes, the
locus is a conic; but when four real tangents can be drawn from a
point on the quartic, care must be taken to select the two pairs
which correspond to the two real or two imaginary tangents which
can be drawn to the lemniscate.
373. Properties of quartics having two nodes and a cusp or
three nodes may be deduced from those of bicircular quartics
having the same singularities. In the first case the bicircular
quartic must be the inverse of a parabola, and in the second case
the inverse of a central conic.
374. Another class of simple forms may be deduced by the
following method, which is one of general application. Let the
triangle of reference be projected into an equilateral triangle in
such a manner that the line la + mw3 + ny = 0 becomes the line at
infinity. Then if (a, 3, y), (a', /', y') be corresponding points in
the two planes, it follows that the line (1, m, n) is projected into
the line a' + f' + y' = 0; hence we may take a' = la, /3' = /3,
y' = n, and the substitution of these values in the equation of the
curve will furnish a simpler curve of the same species.
375. When a trinodal quartic is such that the tangents at
each node together with the lines joining this node to the other
two nodes form a harmonic pencil, the quartic can be projected
into another curve in which the three nodes are situated at the
vertices of an equilateral triangle and the lines bisecting the three
nodal tangents intersect at the centre of gravity of the triangle.
For this special class of quartics the theorems of ~ 192-4 may
be at once proved by inspection; but although theorems connected