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

128

QUARTIC CURVES.

The tangential equation of a conic is
P2+ Q + 2+2pr+ 2q + 2r~. = 0.........(7),
where (,, ) are tangential coordinates. To find the condition
that (5) should touch (7), put - = 0, and (7) becomes
Qq2+ B +2 + 2p7= 0,
whence 1
71?72 - Q -
~+ 2 _2p _21
h1 72 R v2'
by (6). Hence if we take
P/X2 = Q/2 v = -p/l = - q/m = - r/n,
the conic (7) becomes
X2$2 + t2V12 + V22 -21 2b- 2m^ - 2nt = 0......... (8),
which is the equation of a conic touching the six nodal tangents.
By ~ 71 equation (8) when expressed in trilinear coordinates
becomes
(A22 - 12) a2 + (2X2 _ m2) 32 + (X2 2 - n2) y2
+ 2 (mn + X2) + 2 (nl + 2m) a + 2 (Im + v2n) ar = 0...(9).
Equation (9) may also be expressed in the form
v2/s2+vy2 + 21/3y + k2 {(2v2 _ 2) + (m +,2n) /+(ln + 2m) 7}2=0,
where
k2 (X2k2p2 _ 122 m22 - n2v2 - 21mn) = 1,
which shows that the term in brackets is the chord of contact.
The equations of the other chords of contact can be obtained in a
similar manner.
When the nodes are biflecnodes, I= n = n = 0; and the conic
is self-conjugate to the nodal triangle, and becomes identical
with (4).
When the three double points are cusps, I = pv &c., and the
coefficients of a2, /32, 72 vanish. This requires that X21 = mn &c.;
whence the curve becomes
/3y/l + ya/m + al3/n = 0,
which represents a conic circumscribing the nodal triangle.