* See Oh. IV, § 3. Or, if \x + py + vz = and the quartic touch one
another, so do K/x + /i/y + v/s = and ax 2 + 6j/ 2 + cs 2 + 2fyz + 2gex + 2 hay = 0.
Now write down the well-known condition that these two conies Bhould
touch.
XVII 3 TRINODAL QUARTIC 271

For the lines joining the intersections of the line and
quartic to (0, 0, 1) are

6bX 2 x* + 12 (h\ 2 + b\p-f\v)x*y

+ 6(a\ 2 + b/j.' i + ct> 2 ~2fiJLi>-2gi>\+4h\n)x' i y 2

+ 12 (h fi 2, + aX/i- gpv)xy 3 + 6a /i 2 y* = ;

and the cross-ratio (p of this pencil is given by the above
equation (Ch. I, § 11).
Ex. 1. If a quartic with three real nodes has a real bitangent, each
node is either an acnode, or is a crunode such that the tangents at this
crunode and the lines joining it to the other two nodes form two non-
overlapping pairs of lines. In this case all four bitangents are real,
the tangents from the nodes are all real or all unreal, and the quartic
may be put in the form

y i z t + z 1 x 1 + x 2 y'' + 2xyz (fx + gy + hz) = 0.
[For a real bitangent a, b, c have the same sign.]

Ex. 2. A real quartic with three nodes of which only 'one is real has
two real bitangents.
