[459 
459] 
ON THE DOUBLE-SIXERS OF A CUBIC SURFACE. 
327 
and 
The line (a, b, c, f g, h) is given as the intersection of any two of the four planes 
( h, -g, a \x, y, z, w) = 0, 
- K f b 
9> ~f c 
— a, —b, —c, 
or substituting for x, y, z, w the values X + Y+Z —10, Z, —X+Y+Z—10, Y, these 
become 
( 9 > a~9 , 
-f-h, b+f-h, 
9 > c+g , 
c —a, — c — a , 
h-g , 
f-h , 
9-f » 
g \X,Y,Z, 10) = 0, 
h-g 
-9 
c + a 
or, what is the same thing, 
( . , 2g —a+ c , 2g-f-h , -2# 7, 10) = 0. 
— 2g + a — c, . , a + b + c + /— /¿, —a—c 
-2g+f+h, —a—b—c—f+h, . , —f+h 
2g , a + c , -/+A 
And substituting, we have the equations of the several lines, viz.: 
(!') 
X + Y 
= 10, 
£=0, 
(2) 
- X + 7 
= 10, 
z=o, 
(3') 
- X + Z 
= 10, 
7=0, 
(4) 
x+ z 
= 10, 
7=0, 
(S') 
. , 
40, 
5, 
- 4 
№ 
F 2, 10) = 0, 
-40, 
• 3 
55, 
-36 
- o, 
— 55, 
• 3 
1 
4, 
36, 
- 1, 
• 
(6') ( • , 
— 35, 
10, 
- 7 
№ 
F, F 10) = 0, 
-35, 
• 3 
55, 
- 28 
-10, 
-55, 
• 3 
3 
7, 
28, 
- 3, 
.