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Some Properties of a Family of Curves on a Surface

Published online by Cambridge University Press:  20 January 2009

C. E. Weatherburn
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This paper shows briefly how, for a singly infinite family of curves on a given surface, the fundamental properties of the family at any point are associated with three central conies determined by the curves, in a manner resembling that in which the curvature properties of a surface at any point are associated with Dupin's indicatrix. The differential invariants employed are the two-parametric invariants for the given surface.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 1 , Issue 3 , July 1928 , pp. 160 - 165
Copyright
Copyright © Edinburgh Mathematical Society 1928

References

1 Cf. the author's Differential Geometry. Chap. XII ; or Quar. Journ. of Math., 50 (1925), 230–269.Google Scholar

2 Differential Geometry, Art. 116.Google Scholar

3 Cf. The Mathematical Gazette, 13 (Jan. 1926), 2 ; or Diff. Geom., Art. 126.Google Scholar

4 The Math. Gzette, loc. cit., p. 5 ; or Diff. Geom., Art. 125.Google Scholar

5 Diff. Geom., p. 258.Google Scholar

6 Ibid., Art. 121.

7 Ibid., Art. 130.

8 Ibid., p. 221.

9 Diff. Geom., p. 258.Google Scholar

10 Ibid., Art. 130.

11 Ibid., p. 258.

12 Loc. cit.Google Scholar

13 Diff. Geom., p. 250.Google Scholar

14 Neville uses the term “swerve” for a different function. See his Multilinear Functions of Direction, p. 26.Google Scholar

15 “On Curvilinear Congruences,” communicated to the Amer. Math, Soc.Google Scholar