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

TRINODAL QUARTICS.

131

represents any conic which passes through D, D'; E, E'. This
conic may therefore be made to represent (15), in which case we
must have
P =- k1/l, Q =- 2.................. (18),
R + 1/n2 = (2P' + k2/n + /mn)/l = (2Q' + kl/n + 1/ln)/,m
= (2R+' + kc1 + 1/lm)/(lr - n)......(19).
If the conic passes through the two points F, F' it must be
possible to make the equation
FLS ( + k + ) (a/l + 13/n + / ky) = 0......(20)
represent the conic
a2 + (nn - 1) y +m mya + na/3 =0............(21),
which by virtue of (15) is the conic which passes through E, E';
F, F'; B, C. Comparing (20) and (21) we obtain
t= 1, Q=- k/m, R=-k3/n............(22),
P + 1/12 = (2P' + k2k3 + 1/mn)/(mn - 1) = (2Q' + k13/ + l/ln)/m
= (2R' + /1 + l/lm)/n..................(23).
Equations (19) and (23) are six equations for determining
three quantities P', Q', R'; but on solving them it will be found
that they are capable of coexisting, which shows that a conic S
can be described through the six points D, D'; E, E'; F, F'.
The values of P, Q, R are determined by (18) and (22), and by
solving (19) and (23) and taking account of the values of ka, k2, k3
determined by (13), we shall obtain
1 12 +1
2P' =- l - 1 (24),
2P =; - - - -'m.....................(24),
with symmetrical expressions for Q', R'. The conic is therefore
completely determined by (16), (18), (22), and (24).
195. Since a real crunode reduces the number of real points
of inflexion by two and the number of imaginary ones by four,
whilst a conjugate point or an imaginary node reduces the number
of imaginary points of inflexion by six, the number of real and
imaginary points of inflexion of any given trinodal quartic can be
written down. The same can also be done in the case of quartics
having three double points, some of which are cusps.
9-2