410
A MEMOIR ON CUBIC SURFACES.
[412
and therefore
L = y 2 + d.w,
ilf = 2dxy + 2Bw,
X = — 4 d 2 x 2 — 4 Gw,
then we have
P = {— (y 2 + Aw) (4d 2 x 2 + 4<Cw) + (2dxy + 2Biu) 2 }
= — 4 {Gy 2 — 2Bdxy + Ad 2 x 2 + w { AG — B 2 )),
or the equation is
4L 2 {Gy 2 — 2Bdxy + Ad 2 x 2 + w (AG — i? 2 )} + lS^Zil/X + 1 QzM 3 + 27z 2 wN 2 = 0.
101. Consider the section by the plane w = 0, we have L = y 2 , M = 2dxy, iY= — 4<d 2 x 2 ,
and the equation becomes 4y 4 {Gy 2 — 2Bdxy + Ad 2 x 2 ) + (128 — 144 =) — 16d 3 x 3 y 3 z = 0 ; which
substituting for A, B, G the values
A = kx + 12 cz,
B = 6cx — 3 by + Gbdz,
G = 6bdx — 4 ady + 4 ad 2 z,
becomes 16dy 3 (y — dz) (dx 3 — Scx 2 y + Sexy 2 — ay 3 ) = 0 ; which is in fact the line w — 0, y = 0
(reciprocal of the edge) three times, and the lines w= 0, (y — dz) {d, —c, b, — cl$x, y) 3 = 0
(reciprocals of the biplanar rays) each once. Observe that the edge (X = 0, Z = 0) is
not a line of the cubic surface, but the reciprocal line y = 0, w = 0 presents itself as
an oscular line of the reciprocal surface.
102. The equations of the cuspidal curve are in the first instance obtained in the
form
L, M, 3X j = 0.
12 zw, L , — 4 M
Consider the two equations
L 2 — 12zwM — 0,
LM + dzivN = 0,
each of the fourth order, but which are satisfied by zw = 0, L = 0; that is, by
(w = 0, y 2 = 0), (z = 0, y 2 + 4txw — 0). The line (w = 0, y — 0) however presents itself in
the intersection of the two surfaces, not twice only, but 4 times. To show this,
observe that the line in question is a nodal line on the surface L 2 — 1l2ziuM = 0; in
fact, attending only to the terms of the second order in y, w, we find
{{^x + 12cz) 2 — 144cxz — 144bdz 2 } w 2 — 24>dxzyw = 0,
giving the two sheets
{(4x 4- 12oz) 2 — 144cxz — 144bdz 2 } w — 24dxzy = 0 and w— 0 ;
in regard to the last-mentioned sheet the form in the vicinity thereof is given by
w=Ay 3 , viz. we have approximately L = y 2 , M = 2dxy, and thence y 4, — 12z . Ay 3 .2dxy = 0,
that is, A = 24^^ or w = <2Adaz^ ’ ^ ne t ^ lus a fl ecn0( ^ a ^ ^ ne on sur f ace
