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Much has been written, from the algebraical as well as from the geometrical standpoint, on the subject of pencils of quadrics: algebraically the problem consists of the canonical reduction of a pencil of quadratic forms, and the classical paper on the subject is by Weierstrass. But among the different kinds of pencils of quadratic forms there is the “singular pencil,” in which the discriminant of every form belonging to the pencil is zero; interpreted geometrically this means that every quadric belonging to the pencil is a cone. This case was expressly excluded from consideration by Weierstrass, and the canonical reduction was only accomplished later by Kronecker. But, although Weierstrass and Kronecker together solved completely the problem of the canonical reduction of a pencil of quadratic forms, a much clearer insight into the nature of the problem was gained when Segre gave the geometrical solution. He published two papers, one dealing with the non-singular pencils and the other with the singular pencil.
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