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SURFACES, Theory of. Surface, in the mathematical sense, is the common boundary of two contiguous regions of space. The de velopments in this vast field of mathematical investigation are essentially of modern origin. The geometers of the Greek school were ac quainted with some of the elementary proper ties of a few surfaces, notably those of sphere, cylinder and cone, but the systematic and fruit ful study of surfaces began with their repre sentation by means of equations in Cartesian co-ordinates (see GEOMETRY, CARTESIAN). This was not done until the method of co-ordinates had been employed with success in the study of plane curves, whereupon its application to sur faces presented itself as a natural extension. According to Cantor,

In what has been said thus far the point has figured as the primitive element of the surface, and in connection with it the surface is a two dimensional continuum of points. With the

expansion of the subject additional primitive elements were introduced, viz., the plane and the line, and from the standpoint of the new elements the surface may be regarded as a two-aimensional continuum of planes, i.e., ai the envelope of its tangent planes, or as a three dimensional continuum of lines, i.e., the enve lope of its tangent lines. The theory of a sur face as the envelokie of its col of tangent lines constitutes a special chapter in the general theory of complexes of straight lines (see GEOMETRY, LINE, AND ALLIED THEORIES). Along with the analytical method, the synthetic or projective method has been employed, and with special elegance and completeness in the case of surfaces of the second order. With this brief introduction we now pass to a more detailed account of the developments in this branch of mathematics.

1. Algebraic Surfaces in GeneraL—Any surface which can be analytically expressed.by an algebraic equation between the Cartesian co-ordinates x, y, z of a point of space is called an algebraic surface. The -order of the surface is the number of points of intersection .(real or imaginary.) of the surface by an arbitrary straight line. The order of the surface Is obviously the same as the degree n of its equation. The class is the number of tangcnt planes of the surface that pass through an arbitrary line. When there is no singularity (see 7) on the surface the class is n(n— ly. The rank of the surface is the order of a cir cumscribing cone whose vertex is an arbitrary point of space. The rank is n(n —1). The intersection of the surface by a plane is a curve of nth order, and, by the foregoing, the class of this curve is the same as the rank of the surface.

2. The Plane.— This is the simplest of all surfaces, and its equation in the variables x, y, s is of the first degree: Ax By+ az + D in which A, B, C, D are constants. It is the only surface of first order.

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