412
A MEMOIR ON CUBIC SURFACES.
[412
The planes are
The lines are
Z =0,
[10]
X = 0,
Z= 0,
(0)
x = o,
[00]
F = 0,
Z= 0,
(1)
Y + Z= 0,
[12']
X = 0,
F + Z = 0,
(2')
Y-Z = 0,
[13']
X =0,
1
N
II
p
(S')
X-W= 0,
Y + Z — 0,
(12')
x+ w=o,
Y-Z= 0,
(13').
105. The two mere lines are facultative, and the edge is also facultative; p' = b’ = 3 ;
t' = 0.
106. Hessian surface. The equation is
Z(WXZ+ F 2 F + YX 2 -Z S )~ 4>X 2 YZ + X 4 + 4>Z* = 0.
The complete intersection with the surface is thus given by the equations
WXZ+ Y 2 Z + YX 2 -Z s = 0, -4X 2 FF + X 4 + 4X = 0,
which is made up of the line X = 0, Z =0 (the edge) four times and a curve of the
eighth order. To see this, observe that the last-mentioned surfaces have in common
the line X = 0, Z = 0, which is on the first surface a torsal line (equation in vicinity
being Z = -y X 2 ), and on the second surface a triple line (equations in vicinity being
Z=^.X 2 and X 2 =i r Z 3 ). But Z= — ~X 2 touches Z=^.X 2 , and the line counts thus
i Y Y Y
(2 + 2 =) 4 times.
107. I say that the complete intersection is the line (X = 0, Z — 0) three times
together with a spinode curve made up of this same line once and of the curve of
the eighth order; and that thus a = 9.
The discussion of the reciprocal surface in fact shows that the reciprocal of the
edge is a singular line thereof, counting once as a nodal and twice as a cuspidal line
thereof; the cuspidal tangent planes are the reciprocals of the several points of the
edge, and the edge is thus part of the spinode curve. The reasoning may appear to
show that the edge should be counted twice, but it must be counted once only,
making the order = 9 as mentioned.
108. I find that the octic component of the spinode curve is a unicursal curve,
the equations of which may be written
X : Y : Z : TF=160 2 : 40 + 160 5 : 160 :i : -5-80 4 - 160 8 ;
the values of 0 at the binode B 5 are 0=0, 0 = oo, and we thus obtain in the neigh
bourhood thereof the two branches
Z-_5^V Z-WZy and Z-fZy Z__(+')•
W~ ° \ WJ ’ w \w) • w \wj •
