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On the Φ-Invariant of Two Quadrics
Published online by Cambridge University Press: 20 January 2009
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The geometry associated with the five invariants of two quadrics is well known. There appears to be an omission, however, in the treatment of the Φ-invariant. Salmon gives the vanishing of Φ merely as a necessary condition for the possibility of the construction of a tetrahedron self-conjugate for one of the quadrics and having its six edges tangential to the other. Sommerville proves its sufficiency, and shows that this condition is poristic, giving rise to two systems of ∝ tetrahedra of the requisite type. No investigation appears to have been made of the locus of the vertices of the ∝ tetrahedra of each system and the dual problem regarding the nature of the developable surface arising from their ∝ faces.
- Type
- Research Article
- Information
- Proceedings of the Edinburgh Mathematical Society , Volume 10 , Issue 1 , January 1953 , pp. 25 - 36
- Copyright
- Copyright © Edinburgh Mathematical Society 1953
References
page 25 note 1 (1) Salmon, , A Treatise on the Analytic Geometry of Three Dimensions (revised by Rogers), Ch. IX.Google Scholar
(2) Sommerville, , Analytical Geometry of Three Dimensions, Ch. XV.Google Scholar
(3) Turnbull, H. W., “Some Geometrical Interpretations of the Concomitants of Two Quadrics,” Proc. Cambridge Phil. Soc., XIX (1919), 196–206.Google Scholar
page 25 note 2 (1) p. 204, § 201.
page 25 note 3 (2) p. 312, § 15.22.
page 25 note 4 (3) p. 198, §, 5.
page 30 note 1 (4) Sommerville, , Analytical Conies, p. 270, § 23.Google Scholar
page 30 note 2 (4) p. 286, Ex. 2., but the result can easily be proved symbolically.
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