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A MEMOIR ON CUBIC SURFACES. 
[412 
78. The equation of the Hessian surface is found to be 
(yZ + 8 TIO XZW+ Y 2 (yZ - 8 W) 2 + 3 (cX + d Y) XZW + 12 7 SX F- (aX + b Y) 
- ( 7 Z + 8W) (3aX 3 + 9bX 2 Y + 6cXY 2 ) 
— 9X 2 {(ac — b 2 ) X 2 + (ad — be) XY + (bd — c 2 ) F 2 } = 0. 
79. Combining with the foregoing the equation of the surface 
XZW+ Y 2 (yZ+8W) + (a, b, c, d%X, F) 3 = 0, 
it appears that these have along the line X = 0, F = 0 the common tangent plane 
X = 0, or, what is the same thing, that they meet in the line X = 0, F= 0 (the axis) 
twice, and in a residual curve of the tenth order, which is the spinode curve; the 
equations may be presented in the somewhat more simple form 
XZW + F 2 (yZ + 8 TF) + (a, b, c, d\X, Y) 3 = 0, 
-^y8Y 2 ZW-^(yZ+8W)(a, b, c, d\X, F) 3 + 12 7 8XF 2 (aX + bY) 
+ X 4 (— 12ac + 96 2 ) — 3d (4aX 3 F + QbX 2 Y 2 + 4cXF 3 + (¿F 4 ) = 0, 
which, however, still contain the line X = 0, F=() twice. The spinode curve, as just 
mentioned, is of the tenth order; that is, we have er'=10. 
Each of the 6 mere lines is a double tangent to the spinode curve, but the 
transversal is only a single tangent: to show this, observe that the equations of the 
transversal are X = 0, yZ + 8IF + dY — 0 ; substituting in the equations of the curve 
the first equation, that of the cubic surface is of course satisfied identically; for the 
second equation, writing X = 0, this becomes F 2 {— QtyhZW — ^dY(yZ + 8 IF) — 3d 2 Y 2 \ = 0 ; 
or writing herein dY= — (yZ + SW), it becomes F 2 (yZ — 8W) 2 = 0. The value F 2 = 0 
gives X = 0, F=0, 7 i?+8TF=0, viz. this is a point on the axis X = 0, F=0 not 
belonging to the spinode curve; the value (yZ — 8W) 2 = 0 gives a point of contact 
X = 0, yZ + STF+ dY = 0, yZ—8W=0; and the transversal is thus a single tangent. 
Hence the number of contacts is 2.6 + 1, =13; that is, we have /3' =13. 
Reciprocal Surface. 
80. The equation is found by equating to zero the discriminant of the binary 
quartic 
{xX 2 + yXY—(8z + yw)Y 2 } + ^Ziu{X(a, b, c, d^X, F) 3 — 7 SF 4 }, 
or say this is (*$X, F) 4 , where the coefficients are 
Qx 2 + 24 azw, 
3 xy + 13b zw, 
y 2 — 2 (8z + yiu) x + 12c^w, 
— 3 (8z + yw) y + 6dzw, 
6 (8z — yw) 2 .