An Attempt to determine the twenty-seven Lines upon a Surface of the third Order, and to divide such Surfaces into Species in Reference to the Reality of the Lines upon the Surface

Abstract

Preliminary remarks. Contrary to the usual practice I would, in the case of a curve, term singular those points only at which Taylor’s theorem fails for point coordinates, and where in consequence the tangent ceases to be linearly determined; and in like manner term singular those tangents for which the point of contact ceases to be linearly determined. Thus a point of inflexion is not a singular point, but the tangent at such point is a singular tangent. According to the same principle, in the case of a surface, I call singular points those only for which the tangent plane ceases to be linearly determined. I say further that a surface is general as regards order when it has no singular points, general as regards class when it has no singular tangent planes. By class I understand the number of tangent planes which pass through an arbitrary line ; by singular tangent planes, the tangent planes for which the point of contact ceases to be linearly determined. By order of a curve in space, I mean the number of points in which the curve is intersected by an arbitrary plane, by class (as for surfaces) the number of tangent planes (planes containing a tangent of the curve) which pass through an arbitrary point. On account of their reciprocal relation to curves I guard myself from putting developable surfaces on a footing with proper curved surfaces, and call them therefore simply developables without the addition of the word surface, since they do not, like proper surfaces, arise from the double motion of a plane, but arise from the simple motion of a plane.