On the determination of the exterior and interior attractions of ellipsoids of variable densitites. / By George Green, Esq., Caius College.

  • Green, George, 1793-1841
Date:
[1833]
Catalogue details

Licence: Public Domain Mark

Credit: On the determination of the exterior and interior attractions of ellipsoids of variable densitites. / By George Green, Esq., Caius College. Source: Wellcome Collection.

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    XV. On the Determination of the Exterior and Diterior Attractions of Ellijisoids of Variable Densities. By George Green, Esq., Cains College. QRead May 6, 1833.^ The determination of the attractions of ellipsoids, even on the hypo- thesis of a uniform density, has, on account of the utility and difficulty of the problem, engaged the attention of the greatest mathematicians. Its solution, first attempted by Newton, has been improved by the suc- cessive labours of Maclaurin, d’Alembert, Lagrange, Legendre, Laplace, and Ivory. Before presenting a new solution of such a problem, it will naturally be expected that I should explain in some degree the nature of the method to be employed for that end, in the following paper; and this explanation will be the more requisite, because, from a fear of encroaching too much upon the Society’s time, some very comprehensive analytical theorems have been in the first instance given in all their generality. It is well known, that when the attracted point is situated within the ellipsoid, the solution of the problem is comparatively easy, but that from a breach of the law of continuity in the values of the attractions when p passes from the interior of the ellipsoid into the exterior space, the functions by which these attractions are given in the former case will not apply to the latter. As however this violation of the lavl^ of continuity may always be avoided by simply adding a positive quantity, for instance, to that under the radical signs in the original integrals, it seemed probable that some advantage might thus be obtained, and the attractions in both cases, deduced from one common formula which would only require the auxiliary variable u to become evanescent in the final result. The principal advantage how- ever which arises from the introduction of the new variable u, depends VoL. V. Part III. 3F