In this book we regard the theory of a cubic surface as best treated in connexion with a certain figure in space of four dimensions. But the preliminary theorems for a general cubic surface are so intimately related with other results included in this volume that it seems desirable to give some account of them.

The theorem of a double-six of lines. It is convenient to give at once an independent proof of a theorem which arises implicitly below. Suppose that a, b, c, d, e are five lines, no two meeting one another, which have a common transversal, f′. There is then, beside f′, a common transversal of every four of the five lines. The theorem referred to is that the five transversals so arising have themselves a common transversal. Denote by e′ the transversal of a, b, c, d, beside f′, by d′ the transversal of a, b, c, e, beside f′, and so on. We are to prove that the five lines a′, b′, c′, d′, e′ are all met by a line. It is clear that no two of these lines intersect; if, for instance, a′ and b′ were in one plane, every two of c, d, e, all of which meet a′ and b′, would intersect.

We prove that a′, b′, c′, d′, e′ have a common transversal by shewing that the common transversal of a′, b′, c′, d′, other than e, coincides with the common transversal, other than d, of a′, b′, c′, e′.