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

120 QUARTIC CURVES.
nary points of undulation which lie on EF; but since these lines
intersect at the real point
- a = //3 = v/7,
which by virtue of (5) lies on EF, this result is impossible; hence
the relation (14) between the constants is inadmissible, and we
must therefore consider the second equation (13).
Writing k = 44mn, (13) becomes
-^=4X,-mn +4 212.....................(16),
4Xmn 2=2
which by (5) may be expressed in the form
1612m2n2 m2 n2
X v~ = -2 p+ 2..............*..... ( )
Now (10) may be written in the form
(12- ) a2 + 2 + n22 - 2 ( + l6m n7a
9 8mn/ y' 16mn2
-2 (1 162n) a = 0.........(18)
also by means of (16), (13) and (5) it can be shown that
8mn ( 16mn) 16mr) '
accordingly (18) reduces to
{(+ -) a-m3}2+ {(I + a162) a- }2 = 0..(19),
[\ 16mWn 16 mn2
which is the equation of a pair of imaginary straight lines touching the quartic at two imaginary points of undulation which
lie on EF.
183. The equation of the conic S may be written in either of
the forms
S = 'F2 - 4mn37y,
or S = DG2 + 4mnXau/lrv,
and consequently (2) may be written in the form
au2 + DG4= O,