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A MEMOIR ON CUBIC SURFACES.
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Article No. 203. Section XXII = 5(1, 1). Equation X 2 W+Y 2 Z = 0.
203. As this is a scroll there is here no question of the 27 lines and 45 planes;
there is a nodal line X = 0, Y= 0, (5 = 1) and a single directrix line, Z = 0, W = 0.
The Hessian surface is X 2 F 2 = 0; the complete intersection with the cubic surface
is made up of X — 0, F=0 (the nodal line) eight times, and of the lines X = 0, Z = 0,
and F= 0, W = 0, each twice.
The reciprocal surface is x 2 z — y 2 w = 0viz. this is a like scroll, XXII = 5(1, 1);
c' = 0, V = 1.
Article No. 204. Section XXIII = 5(1, 1). Equation X (XW + YZ) + Y 3 = 0.
204. This is also a scroll; there is a nodal line X = 0, F = 0, and a single directrix
line united therewith.
The Hessian surface is A 4 = 0; the complete intersection with the cubic surface is
X = 0, F = 0 (the nodal line) twelve times.
The reciprocal surface is w (xw + yz) — z 3 = 0 ; viz. this is a like scroll, XXIII = 5(1, 1) ;
c' = 0, V = 1.
Annex containing Additional Researches in regard to the case XX = 12— U s ; equation
WX 2 + XZ 2 + Y 3 = 0.
Let the surface be touched by the line (a, b, c, f, g, h), that is, the line the
equations whereof are
( 0, h, — g, a \X, Y, Z, W) = 0.
- h, 0, /, b
9, ~f, 0, c
— a, — b, — c, 0
Writing the equation in the form cW. cX 2 + X (cZ) 2 + c 2 Y 3 = 0, and substituting for
cW, cZ their values in terms of X, F, we have
(_gX +/F) cX 2 + X(aX + bY) 2 + c 2 Y 3 = 0,
that is
( a 2 — eg , 2ab + cf b 2 , c 2 \X, Y) 3 = 0,
or say
(3 (a 2 — eg), 2ab + cf, b 2 , 3c 2 F) 3 = 0,
viz. the condition of contact is obtained by equating to zero the discriminant of the
cubic function. We have thus
27 c 4 (a 2 — eg) 2
+ 4b 6 (a 2 — eg)
+ 4c 2 (2a5 + cf) 3
— 5 4 (2a5 + c/) 2
— 185 2 c 2 (a 2 — eg) (2 ab + cf) = 0,
57—2