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It is known that an algebraic surface in 4-space has four essential characteristics in terms of which all its other characteristics can be expressed. Severi* considers as essential the following : n, the order of the surface ; a, the order of the tangent cone of its projection in a 3-space; /, the number of its apparent triple points; and n', the number of 3-spaces that can be constructed tangent to it and passing through a given point. In order to know anything about a surface in 4-space, it is, then, necessary to know its four essential characteristics. Of course, n may always be assumed. Unless some independent means be found whereby the other three characteristics, a, t, n' can be calculated, we cannot know very much about the surface. It is our purpose in this paper to present a method for the independent determination of these three characteristics. Let F' denote a surface of order n in 4-space, and if the surface is the projection of a surface of the same order in 5-space, let F denote the latter. We introduce three other characteristics of F'°. b, the order of the cone of lines passing through a general point in SA and meeting F' twice; 7, the number of tangent lines of F' passing through a given point; and d, the number of improper double points f on F'. The seven characteristics are connected by the following relations :J

On surfaces in spaces of four and five dimensions