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SynopsisAny non-ruled quartic surface with a double line in space of three dimensions has plane sections of genus 2 and so is a projection of a non-singular duodecimic surface in space of eleven dimensions. This was accepted as an established fact by 1890, but there seems not to be any account of such a projection in action. The following pages are submitted to aid the removal of this century-old anomaly.

SynopsisThe cubic surfaces in, save for the elliptic cone, are, whatever their singularities, projections of del Pezzo's non-singular surface F, of order 9 in. It is explained how, merely by specifying the geometrical relation of the vertex of projection to F, each cubic surface is obtainable “at a stroke”, without using spaces of intermediate dimensions.

The following pages have been written in consequence of reading some paragraphs by Reye, in which he obtains, from a quartic surface, a chain of contravariant quartic envelopes and of covariant quartic loci. This chain is, in general, unending; but Reye at once foresaw the possibility of the quartic surface being such that the chain would be periodic. The only example which he gave of periodicity being realised was that in which the quartic surface was a repeated quadric. It is reasonable to suppose that, had he been able to do so, he would have chosen some surface which had the periodic property without being degenerate; in the present note two such surfaces are signalised.

The six quaternary quartic formswere first obtained by Maschke; it has recently been explained that the quartic surfaces obtained by equating these forms to zero are important constituents of Klein's famous configuration derived from six linear complexes that are mutually in involution. The quartic surface Φi = 0 will be denoted, for each of the six suffixes i, by Mi.

There is, in the second (Cambridge, 1911) edition of Burnside's Theory of Groups of Finite Order, an example on p. 371 which must have aroused the curiosity of many mathematicians; a quartic surface, invariant for a group of 24.5! collineations, appears without any indication of its provenance or any explanation of its remarkable property. The example teases, whether because Burnside, if he obtained the result from elsewhere, gives no reference, or because, if the result is original with him, it is difficult to conjecture the process by which he arrived at it. But the quartic form which, when equated to zero, gives the surface, appears, together with associated forms, in a paper by Maschke1, and it is fitting therefore to call both form and surface by his name.

1. The chief purpose of this paper is to demonstrate the existence of a plane quartic curve with eight undulations, an “undulation” being a point at which the tangent has four-point contact. It is shown that the curvewhere (x, y, z) are homogeneous point-coordinates and f a constant, has undulations at the eight points The curve has, in addition to these undulations, eight inflections which are; in general, distinct. But there are two geometrically different possibilities of their not being distinct, and in either instance they coincide in pairs at four further undulations. Thus two types of curve arise without any ordinary inflections at all, their 24 inflections coinciding in pairs at 12 undulations.

Consider a plane curve C of order n and class X; it is to be supposed throughout that C has only ordinary Plücker singularities, i.e. nodes, cusps, inflections and bitangents. Through any point P1 of C there pass, apart from the tangent at P1 itself, X – 2 lines which touch C; let T12 be the point of contact of any one of these tangents and P2 any one of the n – 3 further intersections of P1T12 with C. Through P2 there pass, apart from the tangent at P2 itself and the line P2P1, X — 3 lines which touch C; let T23 be the point of contact of any one of these with C and P3 any one of its n — 3 further intersections with C.

The following paragraphs have been assembled in consequence of my reading Dr. Cundy’s note on 25-point geometry. Towards the end of it, apparently mindful of the adjunction of a “ line at infinity ” to the Euclidean plane, he adjoins a line to the 25-point plane and so obtains a geometry of 31 points. Here I reverse this procedure : I start with the 31-point geometry and thereafter assign to one of its 31 lines the rôle of the “ line at infinity ”. This seems more in the spirit of Cayley's dictum at the end of his Sixth Memoir on Quantics, that “ descriptive geometry is all geometry ” and metrical geometry only a part thereof.

The pencil of quartic curveswhere x, y, z are homogeneous coordinates in a plane, was encountered by Ciani [Palermo Rendiconli, Vol. 13, 1899] in his search for plane quartic curves that were invariant under harmonic inversions. If x, y, z undergo any permutation the ternary quartic form on the left of (1) is not altered; nor is it altered if any, or all, of x, y, z be multiplied by −1. There thus arises an octahedral group G of ternary collineations for which every curve of the pencil is invariant.Since (1) may also be writtenthe four linesare, as Ciani pointed out, bitangents, at their intersections with the conic C whose equation is x2 + y2 + z2 = 0, to every quartic of the pencil. The 16 base points of the pencil are thus all accounted for—they consist of these eight contacts counted twice—and this set of points must of course be invariant under G. Indeed the 4! collineations of G are precisely those which give rise to the different permutations of the four lines (2), a collineation in a plane being determined when any four non-concurrent lines and the four lines which are to correspond to them are given. The quadrilateral formed by the lines (2) will be called q.