412] 
A MEMOIR ON CUBIC SURFACES. 
387 
51. The six nodal rays are not, the fifteen mere lines are, facultative. Hence 
b' = p' = 15; ¿' = 15. 
52. Resuming the equation W (a, b, c, f, g, h\X, Y, Z) 2 + 2kXYZ = 0, the equation 
of the Hessian surface is found to be 
KW 2 (a, b, c, f, g, h%X, Y, Zf 
4- 2kW{(a, b, c, f g, h\X, Y, Z) 2 (FX + GY + HZ) - 3KXYZ] 
- k 2 {a 2 X 4 + b 2 Y 4 + c 2 Z 4 - 2be Y 2 Z 2 - 2caZ 2 X 2 - 2abX 2 Y 2 
- 4XYZ[(<*/+ gh) X + (bg + hf) Y + (ch +fg) Z]} = 0. 
where 
(A, B, C, F, G, H) = (be —f 2 , ca — g 2 , ab — h 2 , gh — af, hf — bg, fg — ch), 
K = abc — af 2 — bg 2 — ch 2 + 2fgh. 
The Hessian and the cubic intersect in an indecomposable curve, which is the spinode 
curve; that is, spinode curve is a complete intersection 3x4; cr' = 12. 
The equations of the spinode curve may be written in the simplified form 
W(a, b, o, f g, h\X, Y, Z) 2 + 2kXYZ=0, 
- 8KXYZW 
+ 8 kX YZ (afX + bg Y + chZ) 
- k 2 [a 2 X 4 + b 2 Y 4 + c 2 Z 4 - 2bcY 2 Z 2 - 2caZ 2 X 2 - 2abX 2 Y 2 } = 0; 
and it appears hereby that the node C 2 is a sixfold point on the curve, the tangents 
of the curve in fact coinciding with the six rays. 
Each of the 15 lines touches the spinode curve twice; in fact, for the line 12 
we have X = 0, W = 0; and substituting in the equations of the spinode curve, we have 
(bY 2 — cZ 2 ) = 0; that is, we have the two points of contact X = 0, W=0, Y^lb=±Z^Ic. 
Hence /3' = 30. 
Reciprocal Surface. 
53. The equation is found by equating to zero the discriminant of the ternary 
cubic function 
(Xx+ Yy + Zz) (a, b, c, f, g, h\X, Y, Z) 2 — 2kwXYZ, 
viz. the discriminant contains the factor w 2 which is to be thrown out, thus reducing 
the order to n = 10. 
The ternary cubic, multiplying by 3 to avoid fractions, is 
X 3 , Y 3 , Z 3 , 3Y 2 Z , 3ZX , 3X 2 Y , 3YZ 2 , SZX 2 , 3XY 2 , 6XYZ, 
3ax, 3by, 3cz, bz 4- 2fy, cx + 2gz, ay + 2hx, cy + 2fz, az + 2gx, bx + 2hy, fx + gy + hz - ho. 
49—2