Rational Quintic Surfaces with Two Skew Double Lines

The purpose of this paper is to obtain by different methods, and where possible to extend, the results given by Clebsch and Hillf for quintic surfaces having two distinct skew double lines. In a second paper, which will appear in an early issue of this Bulletin, I discuss a surface, due to Montesano, which has two consecutive skew double lines. The equation of a quintic surface that has xy and zw for double lines, but not further specialized, contains twentyfour terms. If we take a point on each of them, say (0:0 : zi : 1) and (xi : 1:0:0) we find that for a given surface there are thirteen pairs of values of Xi and z\ such that the line joining corresponding points lies in the surface. If we allow the coefficients to vary we find that eleven of these transversals may be assigned arbitrarily and the remaining two are then determined. That is, there is a single infinity of such surfaces all of which have the eleven transversals and the other two determined by them. The form of the equation shows that there are six pinch points on each of the double lines. The surface is obviously rational, that is, its points may be put in one to one correspondence with the points of a plane. For the line drawn from any point P of the surface to intersect the two double lines meets a given plane, 7To, in a point P ' , which we may consider the image of P. A section of the quintic surface has of course two double points where its plane cuts the double lines. Therefore, establishing the correspondence in the way just indicated, as P moves on the plane section the surface generated by the