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

134

BICIRCULAR QUARTICS.

where uza, v, are binary quantics in x and y. Equation (4) may
also be written in the form
S2+ =O...........................(5),
where S is a circle and U a conic; or in the form
S2 + U2 = O...........................(6),
where S and U have the same meanings and I is the line at
infinity.
Equation (6) shows that the conic U and the line at infinity
I have a contact of the first order with the quartic at the points
where it is cut by the circle S; and that this circle also has
a contact of the first order with the quartic at the two points
where it intersects the line at infinity. The conic U touches
the quartic at the four points where S and U intersect; but
the contact of the circle and the line at infinity with the quartic
arises from the fact that both pass through the circular points,
which are nodes on the quartic.
199. To find the equation of a cartesian.
The equation
a3 (La + M/3 + Ny) + a2 (X2l2 + 213y + _'2y2)+ 2kvey (X t + vy)
+ k232y2 0 =...............(7)
represents a quartic having a pair of cusps at B and C.
Transform this equation by means of (2) and then put
X + =p, t (\- v)=q,
and it becomes
k2 (X2 + y2)2 2k1 (2 + y2) (pX + qy) + I22 {(p2 - q2) (2 - y2)
+ 2pqxy 2 ( + y2) 3 (LI + Px + Qy)= 0.
Let
p/k== a, Iq/k = b, I2 {2, - 4 (p2 + q2)}/k-2 = 22,
and the equation may be written in the form
(X2 + y2 + ax + by)2 + 2c2 (x2 + y2) + I3 (LI + Px + Qy)/k2 = 0,
which is the same as
(x2 + y2 + ax + by + c2)2 Ax + By+ C = 0.