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

  • George Green
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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    of ii, the function V has the property of satisfying a partial differen- tial equation of the second order, and by an application of the Cal- culus of Variations *it will be proved in the sequel that the required value of V may always be obtained by merely satisfying this equation, and certain other simple conditions when p is equal to the product of two'factors, one of which may be any rational and entire function of the s accented variables, the remaining one being a simple algebraic function whose form continues unchanged, whatever that of the first factor may be. The chief objeqt of the present paper is to resolve the problem in the more extended signification which we have endeavoured to ex- plain in the preceding paragraph, and, as' is by no means unusual, the simplicity of the conclusions corresponds with the generality of the method employed in obtaining them. For when we introduce other variables connected with the original ones by the most simple rela- tions, the rational and entire factor in p still remains rational and entire of the same degree, and may under its altered form be ex- panded in a series of a finite nuinber of similar quantities, to each of which there corresponds a term in V, expressed by the product of two factors; the first being a rational, and entire function of s of the new variables entering into F, and the second a function of the remaining new variable h, whose differential coefficient is an algebraic quantity. Moreover the first is immediately deducible from the corresponding part of p without calculation. The solution of the problem in its extended signification being thus completed, no difficulties can arise in applying it to ,particular cases. We have therefore on the present occasion given two applications only. In the first, which relates to the attractions of ellipsoids, both the interior and exterior ones are comprised in a common formula agreeably to a preceding observation, and the discontinuity before noticed falls upon one of the independent variables, in functions of which both these attractions are expressed; this variable being con- stantly equal to zero so long as the attracted point ^9 remains within the ellipsoid, but becoming equal to a determinate function of the co- 3 F 2
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