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Published online by Cambridge University Press: 20 January 2009
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.
page 259 note 1 “Zur Theorie der bilinearen und quadratischen Formen,” Berliner Monatsberichte (1868), 310–338.Google Scholar
page 259 note 2 “Ueber Schaaren von quadratischen Formen,” Berliner Monatsberichte (1874), 69–76. Kronecker also wrote a later paper on the subject; see the Berliner Hitzungsberichte (1890), 1375, and the continuation of this paper, ibid. (1891), 9 and 33.
page 259 note 3 Memorie Acc. Torino (2), 36 (1884), 3–86.Google Scholar
page 259 note 4 “Ricerche sui fasci di coni quadrici in uno spazio lineare qualunque,” Atti Ace. Torino 19 (1884), 878.Google Scholar
page 259 note 5 An Introduction to the Theory of Canonical Matrices (Blackie, 1932); Ch. 9. The authors solve the more general problem of the canonical reduction of a singular pencil of matrices; when the matrices are symmetric the problem reduces to that of the reduction of a singular pencil of quadratic forms.Google Scholar
page 260 note 1 Journal für Math., 49 (1855), 279–332.Google Scholar
page 260 note 2 Journal für Math., 70 (1869), 212–240.Google Scholar
page 261 note 1 Salmon, Higher Algebra (Dublin 1885), Lesson 19.Google Scholar
page 262 note 1 For these statements cf. Segre: Mem. Ace. Torino (2), 36 (1884), 70. The pencil of quadrics has the characteristic [211 …… 1], the number of l's occurring in the symbol being n−1.Google Scholar
page 262 note 2 Cf. Segre, loc. dt., p. 36.Google Scholar
page 265 note 1 A ruled surface of order N and genus P in [3] has a double curve of order ½(N−l)(N−2)−P, on which there are (N−4){½(N−2)(N−3)−P} triple points. See Edge, The Theory of Ruled Surfaces (Cambridge, 1931), 31, and the references there given.Google Scholar
page 267 note 1 The order of the surface generated by the tangents of a curve of order ν and genus π, in space of any number of dimensions, is 2ν + 2π−2.
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