ON THE DETERMINATION OF THE EXTERIOR AND 

 INTERIOR ATTRACTIONS OF ELLIPSOIDS OF 

 VARIABLE DENSITIES. 



THE determination of the attractions of ellipsoids, even on 

 the hypothesis 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 successive labours of Mac- 

 laurin, 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 follow- 

 ing 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 z 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 vari- 

 able u to become evanescent in the final result. The principal 

 advantage however which arises from the introduction of the new 



