THE GENERAL CUBIC SURFACE; INTRODUCTORY THEOREMS

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 ′.