Proceedings of the London Mathematical Society, 2nd Ser., Vol. 10 (1912), 15-47

I am indebted to Professor C. A. Scott for kindly reading the manuscript of this paper and making some valuable suggestions. The first paper appeared in these *Proceedings*, Vol. 9, p. 51.

Introduction

In continuation of the discussion of Cayley's bilinear space-transformation, those cases are here considered in which the first fundamental sextic degenerates into six straight lines, some of which coincide. The geometrical language of actual and apparent double points ceases to have an obvious meaning ,and more help is derived from the analytical discussion of the numbers of conditions and of independent surfaces, although it is possible to trace the geometrical meaning in some cases by reference to an adjacent position in which the lines do not coincide. if this adjacent non-coincident pair intersect, there is a distinction between the lines in the plane of the pair, which meet both, and other lines which may meet one or neither; so, for the coincident pair which is the limit of this, we expect to find a special plane tangent to all the surfaces at all points of the line, and any line in this plane meets the surfaces in two points on the line. If the adjacent non-coincident pair do not intersect, there is still the distinction between lines meeting both and lines meeting one; the former are the sets of generators of all the quadrics containing the pair of lines. So in the limiting case we expect to find a special family of quadrics touching the surfaces all along the line. But the cubic surfaces can touch each other all along the line without so touching any quadric or plane; in this case the adjacent non-coincident pair, if it existed, would have to be treated as intersecting in —1 points.

The fact that certain cases cannot be derived form cases of non-coincident lines becomes apparent on comparing the diagrams of the second fundamental sextic, given at the end of this paper, with those in the former paper. Some of these involve non-degenerate quartics or pairs of conics, which cannot be derived form any of the former set, which all consisted partly of three straight lines.

If the intersection of two surfaces consists partly of two or more coincident straight lines, the surfaces have contact of first or higher order at every point of that line; analytically, in the general equation of the family, the independent terms or groups of terms, i.e., those by which any two members of the family can differ, are small of the second or higher order in the neighbourhood of the line.