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

POINTS OF UNDULATION.

121

where
2 + = X22 + m) + 22 + + 2 (- mn /
\/j2 = V2/nj= 1l6Xmn
2n2X 2mn2X
+ -- + - a = 0......(20).
Equation (20) will break up into two straight lines if the
coefficient of /3y vanishes; but we have shown that this relation
between the constants is inadmissible. We shall therefore prove
that either of the equivalent equations (16) or (17) resolves (20)
into the product of two linear factors.
Equation (16) reduces the coefficient of f8y to IX/12, from
which it can easily be shown that (20) may be expressed in the
form
/mX X nX Xv 2
- ~ a+ + m - + 8 + - a + ni y - 8n- ) =0,
which represents a pair of imaginary straight lines touching the
quartic at two points of undulation which lie on DG.
184. It thus appears that when the constants are connected
together by the relation (17), the quartic has four imaginary
points of undulation which lie in pairs on the lines EF and DG
respectively. In the same way if
1612m 2n2 12 m2
~V-.X +. 2......................(21),
the quartic will have four more imaginary points of undulation
lying in pairs upon DE and FG. The coordinates of the eight
imaginary points can therefore be found.
Equations (17) and (21) require that 1/X + n/v= 0. If we
take the upper sign it follows from (5) that m = 0, which is
inadmissible. We must therefore take the lower sign, and we
obtain from (5)
21/\ = 2n/v = - m/u,
which by (17) and (21) give
3212mn = - 5X, 16lm2n = 51, 321mn2= - 5v,
which determine X, u, v.
185. The above arrangement of points of undulation is not
the only possible one; for the equation
14a4 + m4/4 - n474 = 0