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The Tritangent Circles of a Circular Quartic Curve

Published online by Cambridge University Press:  20 January 2009

H. W. Richmond
H. W. Richmond
Affiliation:
King's College, Cambridge.
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§1. In a recent paper with this title Prof. W. P. Milne has discussed the properties of the conics which pass through two fixed points of a plane quartic curve and touch the curve at three other points. In dealing with a numerous family of curves such as this it is very desirable to have a scheme of marks or labels to distinguish the different members of the family; Hesse's notation for the double tangents of a C4 illustrates this. By using another line of approach to the subject, by projecting the curve of intersection of a quadric and a cubic surface from a point at which (under exceptional circumstances) the surfaces touch, I find that a fairly simple notation for the 64 conics, in harmony with that for the bitangents, can be obtained. This paper, let it be said, from start to finish is no more than an adaptation of results known for the sextic space-curve referred to; it will be sufficient therefore to state results with short explanations.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 3 , Issue 1 , March 1932 , pp. 46 - 52
Copyright
Copyright © Edinburgh Mathematical Society 1932

References

page 46 note 1 Journal of the London Mathematical Society, 6 (1931), 90.Google Scholar

page 47 note 2 See Proceedings of the Edinburgh Mathematical Society (2), 1 (1927), 31.CrossRefGoogle Scholar

page 48 note 1 Repertorium d. höheren Math. II, 2, 961:Google ScholarEncyklopädie d. Math. Wiss. III, 2, 1407.Google Scholar

page 48 note 2 Bath, F., Journal of the London Mathematical Society, 3, 84.Google Scholar

page 49 note 1 When the plane or quadric passes through T, T is counted as a point of contact. Analogy here would suggest that two tritangent planes or contact-quadrics have become coincident, and the notation bears this out. Or we may visualize two such planes or quadrics touching a small oval on opposite sides and coinciding when the oval has shrunk to nothing.