280 
PROFESSOR CAYLEY ON CUBIC SURFACES. 
96. Writing {a, b, c, dJX, Y) 3 = -<Z(0 2 X-Y)(0 3 X-Y)(fl 4 X-Y), the planes are 
II 
o 
[0] 
ts 
II 
[00] 
o' 
II 
X 
1 
X 
[22'] 
o' 
II 
X 
1 
M 
[33'] 
6 4 X-Y=0, 
[44'] 
d(0 2 X-Y)-Z = O, 
[12] 
<Z(0 3 X- Y)-Z=0, 
[13] 
d(0 4 X-Y)-Z=O, 
[14] 
) 3 -Y(4+9 3 )-W=0, 
C - ' 
CO 
cq 
i i 
? 4 -Y(0 2 +0 4 )-W=O, 
[2'4'] 
XU-Y(*3+4)-W=0, [3'4'] 
97. And the lines are 
a 
b 
c 
/ 
9 
h 
equations may be written 
0 
0 
0 
0 
0 
1 
(0) 
X=0, Y=0 
0 
0 
0 
0 
-1 
d 
( 1 ) 
X=0, cZY + Z=0 
0 
0 
0 
1 
02 
0 
(2) 
0 
II 
N 
0 " 
II 
>4 
1 
X 
0 
0 
0 
1 
0 3 
0 
( 3 ) 
0 3 X— Y=0, Z=0 
0 
0 
0 
1 
04 
0 
( 4 ) 
^ 4 x-y=o, z=o 
-1 
0 
0 
0 
01 
(20 
4 2 x-y=o, 0*X+W=O 
*. 
-1 
0 
0 
0 
01 
( 3 ') 
^ 3 x-y=o, ^x+w=o 
0, 
-1 
0 
0 
0 
01 
(T) 
^ 4 x-y=o, ^X+W=i 
— dO 2 
d 
1 
—(*3+ * 4 ) 
-0304 
d(0 3 O 4 — * 2 * 3 +* 4 ) 
(12 . 3'40 
but for the remaining lines 
—d6 3 
d 
1 
*-(*.+*0 
-0204 
<Z(tfA-4A-K) 
(13 . 2'4') 
(14. 2 , 3 / ) 
the coordinate expressions 
are more convenient. 
-dO, 
d 
1 
-(*,+40 
- 0203 
^(*,* 3 — * A +* 3 ) 
The mere lines are each of them facultative; b'=g'=3; t'= 0. 
98. Hessian surface. The equation is 
{ Z + 3(<?X + <ZY) } { XZ W + Y 2 Z + (a, b, c, dJX, Y) 3 } 
-4Z(a, b, c, dJX, Y) 3 
— 3(4ac— ob 2 , ad, bd, cd, d 2 %X, Y) 4 =0; 
and it is thence easy to see that the complete intersection is made up of the line X=0, 
