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A MEMOIR ON CUBIC SURFACES. 
423 
127. Writing (a, b, c, dQX, Y) 3 = — d (f\X — Y) (/ 2 X — F) (f s X —Y), the planes are 
X=0, [0] 
Z =0, [7] 
W = 0, [ 8] 
f\X - F = 0, [14] 
f,X-Y= 0, [25] 
f 3 X-Y= 0, [36]; 
and the lines are 
X = 0, F = 0, (0) 
/iX — F = 0, Z=0, (1) 
/ 2 X — F = 0, Z=0, (2) 
/sX — F = 0, Z-0, (3) 
/iX — F = 0, W=0, (4) 
f,X-Y= 0, F = 0, (5) 
/3X — F = 0, F = 0, (6). 
128. There is no facultative line; p' = b'= 0, t' = 0; and hence also /3'= 0. 
129. Hessian surface. The equation is 
X {ZW (cX 4- dY) — 3X (ac - b 2 , ad-be, bd-c^X, F) 2 ] = 0, 
so that the Hessian breaks up into the plane X = 0 (axial or common biplane) and 
into a cubic surface. 
The complete intersection of the Hessian with the cubic surface is made up of 
the line X = 0, Y= 0 (the axis) four times; and of a system of four conics, which is 
the spinode curve; d = 8. 
In fact combining the equations 
WXZ+(a, b, c, d$X, F) 3 = 0 
and 
ZW (cX + dF) — 3X (ac — 6 2 , ad-be, bd-c^X, F) 2 = 0, 
these intersect in the axis once, and in a curve of the eighth order which breaks up 
into four conics; for we can from the two equations deduce 
(a, b, c, dQX, Y) 3 (cX + dY) + 3X-(ac — b 2 , ad —be, bd-d^X, F) 2 = 0, 
that is 
(4ac — 3b 2 , ad, bd, cd, d 2 $X, F) 4 = 0, 
a system of four planes each intersecting the cubic XZW + (a, b, c, dQX, F) 3 = 0 in 
the axis and a conic; whence, as above, spinode curve is four conics. 
It is easy to see that the tangent planes along any conic on the surface pass 
through a point, and form therefore a quadric cone; hence in particular the spinode 
torse is made up of the quadric cones which touch the surface along the four conics 
respectively.