We shall denote y + ax, y + bx, ... by a, /?, y, 8, e, £ and
zx 2 + y 3 by U.
Now

2U = (xPy + 8e£
is equivalent to

x 2 {2z — (be + ca + ab + ef+fd + de)y— (ahe + def) x} — 0,
and meets the quartic

U 2 = <xPy8e£
where (a/?y - 8e£) 2 = 0.
Hence

2a — (bc + ca + ab + ef+fd + de)y — (abc+def)x = . (i)
and the nine similar equations are the ten bitangents of the
quartic*

Writing the quartic in the form

(2 z -p 3 x -p 2 y) 2 x 2 + 4 (2a -p 3 x -p t y) y s

= { (4^6 -p*) ® 2 + 3 ( 3 ^ 5 -PiPz) x v + (Apt -p?) y 2 } A

where p 2 , p 3 , p 4 , p 6 , p 6 are the sum of the products of
a, b, c, d, e, f two, three, four, five, six at a time, we see that
the tangent at /(= J) is

2z-p 3 x-p 2 y = 0,
and that the points H, K of § 1 are its intersections with

(*Pb -2% 2 ) * 2 + 3 ( 3 F5 -P2P3) X V + ( 4 ^4 -Pi) y 2 = 0.
Some of the properties of the cuspidal quartic may be
verified on Fig. 2. In it a, /3, . . . denote the points of contact

of oc = 0, j8 = They are the points with abscissae

— £, — 1, — -|, f , I, 5 ; the cusp being (0,.co ), and the cuspidal
tangent x = 0.
