The collected mathematical papers of Arthur Cayley.

-436

A MEMOIR ON CUBIC SURFACES.

[412

The section by the plane w= 0 (reciprocal of B3) is w=0, y= 0 (the edge) three
times; and w=O, y-x=0; w= 0, y-z=O (reciprocals of the GB-axes).
163. Nodal curve. This is the line y=x=z; wherefore b'= 1, To put the line
in evidence, write for a moment x = y + a, z = y + ry, then the equation is readily converted into
w*. 16 (a - 7)2
+w2 - y (a2-4a7 +72) 1)
+ (a +7y)(2a -7)(-a+2y)
+W 2 y2(a2- 1Oa7y+72)(
- 18ya7y(a + y)
-27a2y7 I
+ y3aoy 0,
which, each term being at least of the second order in a, y (= -y, z -y), exhibits the
nodal line in question.
164. Cuspidal curve. Multiplying by 27, the equation may be written
(7y - 3x - 3z - 5w, - y + 6w, - wy2 + 16yw - 12xw - 12zw + 16w2,
-20y2 yx+ 24 y + 2y- 27xz - 8yw + 16w2)2 = 0,
where
4w (7y - 3x - 3z - 5w) + (- y + 6w)2 = y2 + 16yw - 12 (x + z) w + 16w2;
and we have thus in evidence the cuspidal curve,
y2 + 16yw - 12 (x + z) w + 16w2 = 0,
- 20y2 + 24y (x + z) - 27xz - 8yw + 16w2 = 0,
which is a complete intersection, 2 x 2, or quadriquadric curve; c' = 4.