Surface

The class of f is the degree of its equation regarded as an envelope, connecting the tangential coordinates of any one of its tangent planes. The properties of the class are dual to those of the degree ; it is the class of the tangent cone drawn to f from a general point and the number of tangent planes through a general line.

The rank is an intermediate numerical characteristic, the num ber of tangent lines that can be drawn through a given point and in a given plane through that point ; it is the class of the plane section, and the degree of the tangent cone.

If the polynomial f falls into rational factors f2, of grees n1, where then each factor by itself represents a distinct algebraic surface, and if a point lies on either, its co ordinates satisfy the equation of f ; regarded still as a single sur face, f is degenerate, breaking up into the components fi, f2.

Polars.

The point dividing the segment in the ratio X; I has four homogeneous co-ordinates of the form xX =x-1-74. The condition that Px lies on f is that X satisfies The surface A' for r = 1, • • • , 7/— I, is the rth polar surface of with regard to f ; the first polar passes through all the mul tiple points of f. In particular, or is its polar plane, and its polar quadric. The locus of points whose polar quadrics are cones is the Hessian of f. Its equation of degree 4(n— 2) expresses that the determinant of second derived functions of f vanishes.

If two roots of F=o are infinite, then lies on f, and P on the tangent plane at the two conditions are = o. The lat ter represents the tangent plane at which is the polar plane of its point of contact. If is a multiple point, the tangent plane does not exist, and all the first derived functions of f vanish there.

If two roots of F = o are equal and finite, its discriminant vanishes. This equation of degree in x, y, z, w is satisfied if touches f at Px; in general it represents the tangent cone from to f. Hence the class of f cannot exceed It is less than this if f has multiple curves; for the condition is satisfied if Px is a multiple point of f, and the cone vertex standing on a multiple curve separates itself from the tangent cone.

If two roots of F = o are o, then touches f at P, and f = o, Aof =0. The curve of contact of the tangent cone drawn from any point to f is the whole or part of the intersection of f with the first polar of any residual intersection consisting of the multiple curves of f. This curve of proper contact passes through all the isolated multiple points of f.

Indicatrix; Curvature.

In order to examine a surface in the neighbourhood of a particular point, we take this as origin. 0 of rectangular cartesian coordinates, and arrange f in homo geneous functions of x, y, z, say, f=uo-Hil+ . . . 1-un. If f con sists of a single homogeneous function u,, it represents a cone vertex 0 of degree n. If f is general, the condition that 0 lies on the surface is = o.

If P (x, y, z) is a general point of space near 0, at a distance small of first order, then the perpendicular distances of P from all general planes through 0, viz.: x, y, z and all general linear functions of them, are small of first order. If P lies on f, the linear function u1, being equal to — (u2+ . . . +24), is small of second order ; = o is the equation of the tangent plane, which in the neighbourhood of 0 lies infinitely closer to f than any other plane. Whenever we can safely neglect small quantities of second and higher order, the surface can be replaced by its tangent plane.

If we neglect terms of third and higher orders only, f can be replaced by the quadric or more generally by any of the family q----u1+742-Euivi, where is a general linear function. These are the osculating quadrics of f at 0, any one of which is a second approximation to the surface. The sections of f and q by any plane through 0 have the same curvature at 0.

Let 0 be an ordinary point of f, with z as tangent plane, and therefore x= y = o as normal. Then • • • and the simplest osculating quadric is v2, where v2 is the result of putting z= o in u2.

A plane z= a small constant parallel and close to the tangent plane, cuts f approximately in the conic = with centre on the normal, called the indicatrix.