XV. Oil the Determination of the Exterior and Interior Attractions of 

 Ellipsoids of Variable Densities. By George Green, Esq., 

 Caius College. 



[Read 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 p 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 law of continuity may always be avoided by simply adding a 

 positive quantity, u" 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. SF 



