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The reciprocation of one quadric into another

Published online by Cambridge University Press:  24 October 2008

H. F. Baker
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It is a familiar fact that if two quadric forms, in (n + 1) homogeneous variables, be each expressible as a sum of squares, of (n + 1) independent linear functions of the variables, then they are polar reciprocals of one another, in regard to any one of 2n quadrics. The question arises whether this is true for any two quadrics. Segre states that this is an unsettled question. A solution is given, however, by Terracini for any two non-degenerate quadrics, supposed to have been reduced to the Weierstrass canonical form. The present note has the purpose of pointing out that a solution is derivable from a remark made by Frobenius; this requires a knowledge of the roots of the equation satisfied by the matrix of the two quadrics.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 23 , Issue 1 , February 1926 , pp. 22 - 27
Copyright
Copyright © Cambridge Philosophical Society 1926

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References

* Encykl. Math. III, C. 7, p. 864, 1912.Google Scholar

† Ann. d. Mat. XXX, 1921, p. 155.Google Scholar

‡ Berlin. Sitzungsber. 1896, p. 7Google Scholar. Frobenius refers to Kelland, and Tait, , Quaternions (1873), chap. X, and to Sylvester, Papers, III, pp. 562–567 (1882). Sylvester refers to Babbage's Calculus of Functions.Google Scholar

§ This equation may be the same for cases of different invariant factors. For example, if m be the matrix of the coefficients of the form 2xy+z 2+t 2, and M 1, M 2 be the respective matrices of the coefficients of the two forms

both the matrices M 1m, M 2m satisfy the equation

* Proc. Lond. Math. Soc. XXX, 1899, p. 196.Google Scholar