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On the configuration known as a double-six of lines

Published online by Cambridge University Press:  31 October 2008

H. W. Richmond
Affiliation:
King's College, Cambridge.
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In geometry of three dimensions it is well known that, when two quadrics Q1, Q2 are given, if one set of four points exists having the properties that each point lies on Q1, and each two points are conjugate with respect to Q2, an infinity of such sets of points can be found. The quadrics Q1Q2 stand in a special relation to one another, expressed by the vanishing of the coefficient of A in the discriminant of Q2 + λQ1 an invariant of Q1, Q2. Two quadrics Q1, Q2 are thus related if the equation of Q1 contains no squares of the coordinates, and that of Q2 contains no products of two coordinates; for then the vertices of the tetrahedron of reference form such a set of four points.

Type
Research Article
Copyright
Copyright © Edinburgh Mathematical Society 1937

References

1 See Salmon, , Analytic Geometry of Three Dimensions, Rogers' revised edition, Vol. 30, p. 204Google Scholar; or Sommerville, , Analytical Geometry of Three Dimensions, p. 309.Google Scholar