If f is of degree n, it has ) (n+2)(n+3) —1 inde pendent ratios of coefficients ; all such surfaces form a linear family of N degrees of freedom, and one can be made to pass through N points in general position, each of which presents one independent condition to f. Thus a quadric can be drawn through nine points, and these determine it if they are general. However, seven points determine a net of quadrics, whose base consists of eight points, viz., the seven and another, associated with and determined by the seven, which offers no independent condition; but the passage of the quadric through it is a necessary conse quence of its passage through the others. Thus nine points do not determine a quadric if any eight of them are an associated set of this nature.
To have a given point as a double point imposes four conditions on f ; to contain a given line, n+ i conditions. The number of independent conditions presented to f by an assigned element of any sort is the postulation of that element. The conditions for a combination of elements may force f to contain other fixed ele ments, or to break up. If six out of nine points lie on a plane, the only quadric containing them all consists of this plane and the plane of the other three points.
Two surfaces are orthogonal if at every common point the angle between them, i.e., between their normals, is a right-angle. If three surfaces are mutually orthogonal, their curves of inter section are curves of curvature on each surface. If three linear families are such that the three surfaces through any point are at right angles to each other, they form an orthogonal system; f can form part of such a system only if it satisfies a certain differential equation of third order. In the same way, a cubic family, with three members through any point, can be an orthogonal system, e.g., the confocal quadrics Among the infinite number of surfaces having the same boun dary, i.e., a given closed curve through which each is to pass, the minimal surface is that whose area is least. This is the form assumed by a soap film bounded by a wire in the form of the given contour. The same property must attach to any small portion of the surface, and leads to a differential equation of second order, the boundary conditions determining the arbitrary elements of the solution. A minimal surface is anticlastic everywhere. If it were synclastic at a point P, a plane near the tangent plane at P on one side would cut it in a small plane closed curve, approximately the elliptic indicatrix, whose area would be less than that of the cap on which P lies; and the total area could be reduced without affecting the given boundary.
In general, 4), x are many-valued functions; if there are r sets of values of x, y, z for each set of u,v, and we take u.v. as coordinates of a point of a plane, then f is represented on this plane by an r, I point transformation. The nature of the repre sentation depends largely on the form of the expression for the element ds of length of arc traced on the surface, given by where E, F, G are certain functions of u, v. Here El du, GI dv are the elements of length along the curves represented by v = const., v = const., at the point whose parameters are u,v; while F depends also on the angle between these curves. If they are every where at right-angles, e.g., if they are the curves of curvature, then F=o.
Rational surfaces are those which can be represented on a plane by a 1, z point transformation. Then the coordinates of any point on f are rational functions of two parameters, the coordinates of the corresponding point of the plane. Of the different plane representations of f, the lowest is that for which the curves in the plane corresponding to the plane sections of f are of lowest degree. Any two of these have n variable intersections, corresponding to the points of f on the common line of the two planes of section ; the remaining intersections are fixed, at points of the plane funda mental for the representation. An important family of curves on f are those corresponding to the lines of the plane. Any two of these have one and only one variable intersection, the remainder falling at points of f fundamental for the representation. All quadrics and cubics are rational. Surfaces of higher degree are not, unless they have a sufficient number of singular points and curves.
There are several numerical characteristics of a surface that vanish if it is rational, and may be regarded as answering to the genus of a curve. The simplest is the genus as defined by Noether (Fldchengeschlecht), the number of linearly independent adjoint surfaces of degree 15— 4, passing s— I times through every ordinary s-f old curve of f, and times through every ordinary isolated s-f old point, with other conditions at extraordinary singularities. It is also the number of linearly independent double integrals of the form ff F ( x ,y ,z ) dxdy, where F is a rational function, and x,y,z are connected by f =o. If f has no singularities, its genus is i(n—i)(11-2)(n-3).
(H. P. Hu.)