PEOFESSOE CAYLEY ON CUBIC SUEFACES. 
259 
51. The six nodal rays are not, the fifteen mere lines are facultative. Hence 
b'=tf= 15; £'=15. 
52. Resuming the equation W (a, b , c,f g, Aj^X, Y, Z) 2 4-2EXYZ=0, the equation 
of the Hessian surface is found to be 
KW 2 0, b, c,f, g, hJX, Y, Z) 2 
+2 7cW{(a, b , c,f, g, hJX, Y, Z) 2 (FX+GY+HZ)-3KXYZ} 
~/F{« 2 X 4 + & 2 Y 4 + c 2 Z 4 - 2hcY 2 Z 2 - 2caZ 2 X 2 - 2 «5X 2 Y 2 
- 4XYZ[(^+ ? 4)X+.(5?.+A/)T+(«&+/y)Z] } = 0, 
where 
(A, B, C, F, G, H) = (fo-/ 2 , m-# 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 3x1; <f=12. 
The equations of the spinode curve may be written in the simplified form 
W(a, b, c,f ; g, hJX, Y, Z) 2 +2/rXYZ=0, 
-8KXYZW 
+ SJcXYZ(afX+bgY + chZ) 
— ~k 2 { a 2 X 4 + Z» 2 Y 4 + c 2 Z 4 — 2 5cY 2 Z 2 — 2 mZ 2 X 2 — 2«5X 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 — <?Z 2 ) = 0 ; that is, we have the two points of contact X=0, W=Q, Y\Z6=±Z/y/c. 
Hence /3'=3Q. 
Reciprocal Surface. 
53. The equation is found by equating to zero the discriminant of the ternary cubic 
function 
(K.x J r Yy-\- Zz)(a, b, c,f g, hJX, Y, Zf-2JcwXYZ, 
viz. the discriminant contains the factor w 2 which is to be thrown out, thus reducing 
the order to w'=10. 
The ternary cubic, multiplying by 3 to avoid fractions, is 
X 3 , Y 3 , Z 3 , 3Y 2 Z , 3Z 2 X , 3X 2 Y , 3YZ 2 , 3ZX 2 , 3XY 2 , 6XYZ, 
3 ax, 3 by, 3 cz, bz-\-2fy , cx-{-2gz, ay-\-2hx, cy-\-2fz , az-\-2gx, bx-\-2hy, fx-\-gy-\-hz—Jcw. 
Write as before (A, B, C, F, G, H) for the inverse coefficients (A—bc—f 2 , &c.), 
and K = abc — af 2 — bg 2 — ch 2 + 2 fgh ; and moreover 
*=(A, B, C, F, G, H**, y, z)\ 
P =A^ + H?/+Gr2, 
2n2 
