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Polarities for the nodes of a Segre cubic primal in space of four dimensions

Published online by Cambridge University Press:  24 October 2008

H. F. Baker
H. F. Baker
Affiliation:
St John's College
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Extract

The present note was inspired by the desire to see whether the exposition of the theory of the Segre cubic primal of ten nodes was simplified by using only five coordinates instead of the six redundant coordinates introduced by Stéphanos and Castelnuovo. But the simple remark that the ten nodes may be separated into two simplexes which are polars of one another in regard to a quadric—which may or may not be novel—suggests the comparison of the theory of five associated lines in space of four dimensions with familiar properties of eight associated points in three dimensions; especially as it appears (§ 8) that the ten nodes do not form an associated set. With the repetition, for the sake of clearness, of several results which are familiar in general terms, there seems enough novelty to make the note of some utility. The form found for the equation of the Segre primal in five coordinates (§ 5) seems also noticeable.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 32 , Issue 4 , December 1936 , pp. 507 - 520
Copyright
Copyright © Cambridge Philosophical Society 1936

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References

* See a paper by Thomas Weddle, F.R.A.S., Mathematical Master of the National Society's Training College, Battersea, , “On the theorems in space analogous to those of Pascal and Brianchon in a plane, Part II”, Cambridge and Dublin Math. Journal, 5 (Vol. 9 of the Cambridge Math. Journal) (1850), 58Google Scholar. Among other things he states the result: Let 8 planes intersect, three and three in order, in 8 points. If the opposite planes intersect in 4 straight lines belonging to the same system of generators, in an hyperboloid of one sheet, every surface of the second degree passing through 7 of the points will pass through the eighth.

* This is the point of view of the original paper by Stéphanos, , Compt. rend. 93 (1881), 634 and 578Google Scholar. See also Castelnuovo, , Atti Veneto, 2 (1891), 879.Google Scholar