408 A MEMOIR ON CUBIC SURFACES. [412 
96. Writing (a, b, c, d^X, Y) 3 = —d(0,X—Y)(0 3 X—Y)(0 i X—Y), the planes are 
x = o, 
[0] 
Z = 0, 
[00] 
0 % X- F = 0, 
[221 
0 3 X - F=0, 
[3T] 
o' 
II 
1 
Tf 
[441 
yti 
N 
1 
N 
1 
N 
II 
[12] 
d(0 3 X-Y)-Z = 0, 
[13] 
d(0 i X-Y)-Z = 0, 
[14] 
X0A-Y (0 2 + 0 3 )— W—0, 
[2'31 
X0A-Y(0 2 + 0,)-W = O, 
[2 / 4 / ] 
X0A- Y(0 S + 0i) — W = 0, 
[3'41 
97. And the lines are 
a 
b 
c 
/ 
9 
h 
equations may be written 
0 
0 
0 
0 
0 
1 
(0) X=0, 7=0 
0 
0 
0 
0 
-1 
d 
(1) X= 0, dY + Z= 0 
0 
0 
0 
1 
0* 
0 
(2) 0,X - Y = 0, Z=0 
0 
0 
0 
1 
0-s 
0 
co 
h 
1 
II 
© 
N 
II 
o 
0 
0 
0 
1 
04 
0 
(4) 0,X - Y = 0, Z = 0 
0 a 
-1 
0 
0 
0 
0i 
(2') 0 2 X - 7 = 0, 0 2 X + W = 0 
^3 
-1 
0 
0 
0 
03 2 
(3') 0 3 X -7=0, 0 2 X + TF= 0 
0i 
-1 
0 
0 
0 
0? 
(4') 0,X - 7=0, 0 2 X + TF = 0 
1 - dO, 
d 
1 
— (03 + 0 4 ) 
— 0 3 0 4 
d (0 3 0 A - 0 2 0 3 - 0 2 0i) 
(12. 3'4') , „ , . . 
but tor the remaining lines 
- d6. 
d 
1 
— (02 + 0i) 
-0A 
d(6A-»A-<W 
(13 . 2'4') the coordinate expressions 
are more convenient. 
- d0\ 
d 
1 
— (0 2 + 6 3 ) 
-0A 
d (0 2 0 3 - 0A- 0A) 
(14 . J) 
The mere lines are each of them facultative ; b' = p = 3 ; t' = 0. 
98. Hessian surface. The equation is 
{Z+S(cX + dY)}{XZW+ Y 2 Z + (a, b, c, d\X, Y) 3 } 
— 4Z (a, b, c, d][X, Y) 3 
— 3 (4ac — 3b 2 , ad, bd, cd, d 2 \X, F) 4 = 0 ; 
and it is thence easy to see that the complete intersection is made up of the line 
X = 0, F= 0 (the axis) three times, and of a curve of the ninth order, which is the 
spinode curve; a = 9.