Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-28T12:44:26.507Z Has data issue: false hasContentIssue false
  • English
  • Français

Solutions of Laplace's equation in an n-dimensional space of constant curvature

Published online by Cambridge University Press:  20 January 2009

H. S. Ruse
H. S. Ruse
Affiliation:
University College, Southampton.
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This paper is a sequel to an earlier one containing a tensor formulation and generalisation of well-known solutions of Laplace's equation and of the classical wave-equation. The partial differential equation considered was

where is the Christoffel symbol of the second kind, and the work was restricted to the case in which the associated line-element

was that of an n-dimensional flat space. It is shown below that similar solutions exist for any n-dimensional space of constant positive or negative curvature K.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 6 , Issue 1 , May 1939 , pp. 24 - 45
Copyright
Copyright © Edinburgh Mathematical Society 1939

References

page 24 note 1 Proc. Edinburgh Math. Soc. (2), 2 (1930–1931), 181.Google Scholar

page 26 note 1 Ruse, , Proc. London Math. Soc., 32 (1931), 90.Google Scholar

page 26 note 2 See the paper quoted above and also Ruse, , Quarterly J. of Math. (Oxford), 3 (1932), 15.Google Scholar

page 26 note 3 Synge, , Proc. London Math. Soc, 32 (1931), 241.CrossRefGoogle Scholar

page 26 note 4 Ruse, , Quarterly J. of Math. (Oxford), 1 (1930), 146.Google Scholar

page 26 note 5 Etherington, , Phil. Mag., 15 (1933), 761.CrossRefGoogle Scholar

page 26 note 6 Darboux, , Lecons sur la theorie gene'rale des surfaces, Vol. II, Book V, Ch. V (p. 450, § 536 in 2nd edition, Paris 1915).Google Scholar

page 28 note 1 Ruse, , Quarterly J. of Math. (Oxford), 1 (1930), 148;Google ScholarcfDuschek-Mayer, , Lehrbuchder Differentialgeometrie, Vol. II (Leipzig, 1930), p. 143.Google Scholar

page 33 note 1 A treatise on differential equations (5th ed., 1921), p. 534, § 276.Google Scholar

page 35 note 1 Darboux, , op. cit., Vol. III, Book VII, Ch. XI (p. 394, § 782 in 1st edition, 1894).Google Scholar

page 35 note 2 CfDarboux, , op. cit., Vol. III, p. 396, §783, formula (6).Google Scholar

page 35 note 3 Euler, , Opera omnia, 1st series, Vol. 13 (Leipzig 1913), p. 212et seq.;Google Scholar also Oeuvres de Laplace, Vol. IX (Paris 1893), p. 41,Google Scholar § IX ; and Darboux, , op. cit., Vol. II, Book IV, Ch. III (p. 54 in 2nd ed.).Google Scholar

page 36 note 1 Darboux, , op. cit., Vol. II, p. 65, equation (26).Google Scholar

page 36 note 2 The £ of (4.3) are of course different from the £ of (3.15).

page 38 note 1 We are assuming h > 0 for the sake of the argument. If is the timeof dispatch and the time of reception.

page 39 note 1 Ruse, , “On the definition of spatial distance in general relativity,” Proc. Roy. Soc., Edinburgh, 52 (1931–2), 183–194, and in particular p. 193. For other workon the definition of spatial distanceCrossRefGoogle Scholar see Whittaker, E. T., Proc. Roy. Soc. (A) 133 (1931),93;CrossRefGoogle ScholarKermack, M'Crea, and Whittaker, , Proc. Roy. Soc, Edinburgh, 53 (1932-3), 35;Google ScholarRuse, , Proc. Roy. Soc, Edinburgh, 53 (1932-3), 79;CrossRefGoogle ScholarEtherington, , loc cit. supra;Google ScholarWalker, , Quarterly J. of Math. (Oxford), 4 (1933), 71 ; Temple, Proc Roy. Soc. (A) 168 (1938), 136.Google Scholar