398 
A MEMOIR ON CUBIC SURFACES. 
[412 
74. Writing X (a, b, c, d$X, Yf-ySY* = - 7 3 (/ 4 X - F)(/ 2 X- Y)(f 3 X - F)(/ 4 X - Y), 
planes are 
X = 0, 
[ o] 
X —f 1 Y= 0, 
[IF] 
X —f,Y= 0, 
[22'] 
X-/ 3 F= 0, 
[33'] 
X-f i Y = 0, 
[44'] 
=0, 
[12] 
8{X-(f 1 + f 3 )F}-f 1 f 3 Z =0, 
[13] 
S{X-(f 1 + f 4 )F}-f 1 f 4 Z =0, 
[14] 
8{X-(f 2 + f 3 )F}-f 2 f 3 £ = 0, 
[23] 
3 {X - (f 2 + f 4 ) F} - f 2 f 4 Z = 0, 
[24] 
8{X-(f 3 + f 4 )F}-f 3 f 4 £ =0, 
[34] 
7 {X — (fi + f 2 ) F}-f x f 2 TF=0, 
[12'] 
7 {X-(f 1 + f 3 ) F} — fjf 3 Tf = 0, 
[in 
7 {X-(f 1 + f 4 )F}-f 1 f 4 F=0, 
[in 
7 {X — (f 2 + Q F} — f 2 f 3 W = 0, 
[2'3'] 
7 {X-(f 2 + f 4 ) F} — f 2 f 4 W = 0, 
[2'4'] 
7 {X — (f 3 + f 4 ) F} — f 3 f 4 TF = 0, 
[3'4'] 
— 7 s ^ ^ + f? ) X + cfo/ + 7 ^T + 8 IF = 0, 
[12.34] 
— (f7 f~f) ^ + dy + yZ + 8 F = 0, 
[13.24] 
-7S (fi + -) X + cfy + 7 Z+81F = 0, 
[14.23] 
And the 16 lines are 
(a) 
(P) 
(c) 
(/) 
(9) 
(h) 
whence equations may be written 
0 
0 
0 
0 
0 
1 
o 
II 
o 
II 
N 
S 
8 
0 
0 
0 
- 7 
d 
(5) X=0, dY + yZ + 8W = 0 
0 
0 
0 
fa 2 
fi 
-8 
(1) X- f 4 F = 0, SF + i x Z— 0 
0 
0 
0 
f., 2 
f. 
-8 
(2) 
0 
0 
0 
fa 2 
fa 
-8 
(3) 
0 
0 
0 
f/ 
fa 
-8 
(4) ii ii