GENERAL EQUATION OF A CUBIC SURFACE. 
47 
1 4 7 10 
1 5 8 12 
1 6 7 12 
2 5 8 11 
2 5 9 10 
2 6 7 11 
3 4 8 11 
3 4 9 10 
3 6 9 12. 
§ 4. In these groups there are distinct types of relation. 
Each of the three groups 
1 6 7 12 
2 5 8 11 
3 4 9 10 
represents two pairs of intersecting lines ; for instance, the pair 3 and 10 intersect 
each other, and the pair 4 and 9 intersect each other, but neither 3 nor 10 intersects 
4 or 9. 
It is clear that the intersection of the plane containing the lines 3 and 10 and the 
plane containing the lines 4 and 9 meets the surface in four points, and therefore lies 
entirely on the surface. 
Its equations are 
- + v = T 
ft d 
H13). 
I 
J 
In the same way it follows that the intersection of the planes of the lines 2, 8, 
and 5, 11, is a line on the surface, whose equations are 
]L \ ^ 
h d 
T + T 
T 
1 
H14). 
= T 
J 
and that the intersection of the planes of the lines 1, 6, and 7, 12, is a line on the 
surface, whose equations are 
