Some Considerations on the General theory of Ruled Surfaces
Proceedings of the London Mathematical Society
1919.] THE GENERAL THEORY OF RULED SURFACES. 281 and having the same moduli, the lines joining the corresponding points of a (1, 1)-correspondence between them generate a ruled surface of order m-\-n. If the curves have k self-corresponding points, the order of the surface is diminished by k. Any plane passing through a generator of a scroll Ff L * meets the surface in a residual curve of order n-1 met by the generator in n-1 points. Each of these is in a sense a double point for the plane
... for the plane section, but only one of them is the point of contact of the plane with the surface. The remaining n-2 points are fixed points on the generator not varying as the plane moves about the line. Fort consider any two planes through the generator. Each of these contains a residual C n -\. These two curves are placed in (1, ^-correspondence by the generators, and the generators constitute a scroll of order n. Hence in virtue of the result cited above, the two curves must have n-2 common self-corresponding points. But the common points of the two curves must lie on the line common to the two planes, and that is the generator. These are then n-2 fixed points on the generator, not varying with the plane. Through each of these n-2 points there passes another generator, and they are points of a double curve on the surface. The double curve is, in fact, the locus of points on the scroll through which pass two generators. A scroll Fn has thus in general a double curve met by every generator in n-2 points. This singular curve on the scroll may be of higher multiplicity, and it may also split up into distinct and separate curves each with its own order of multiplicity. Since any scroll must have at least a double curve,! it follows on reciprocation that for any scroll there exists in general an infinity of planes passing through two generators. Hence, finally, the upper limit to the number of generators which may lie in a plane depends on the multiplicity of the singular curve of the reciprocal scroll, and the multiplicity of any multiple points that curve may possess. 3. The equations to any scroll may be expressed in the form x = a+pz, y = b+qz, (\, /J) = 0, where a, b, p, q are algebraic single-valued functions of two variables A, u and represents a curve of the same order and with the same moduli as the plane section of the scroll. The generators are thus in (1, 1)-correspondence with the generators of a cone of the same order; the points of • We indicate a scroll of order n and genusp by Ff t ; and so also a curve by C«. t This remark is due to Oayley, Collected Papers, Vol. 2, p. 33. $ We are just now concerned only with scrolls in ordinary space.