Under the present form it is clear the determination of V can offer no difficulties after what has been shown (Art. 2). I shall not therefore insist upon it here more particularly, as it is my intention in a future paper to give a general and purely analytical method of finding the value of V, whether p is situated within the ellipsoid or not. I shall therefore only observe, that for the particular value

the series U'+ U' + U' + &c. (Art. 2) will reduce itself to the single term U, and we shall ultimately get

1-n

2

ƒ ̃d0'sin 0'ƒ2*da' (a2cos 02+b2sin@cos"+c2sin @sina"),

which is evidently a constant quantity. Hence it follows that the expression (30) gives the value of p when the fluid is in equilibrium within the ellipsoid, and free from all extraneous action. Moreover, this value is subject, when n<2, to modifications similar to those of the analogous value for the sphere (Art. 7).

ON THE DETERMINATION

OF THE

EXTERIOR AND INTERIOR

ATTRACTIONS OF ELLIPSOIDS

OF

VARIABLE DENSITIES.

* From the Transactions of the Cambridge Philosophical Society, 1835.

[Read May 6, 1833.]

ON THE DETERMINATION OF THE EXTERIOR AND ELLIPSOIDS OF

INTERIOR

ATTRACTIONS 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 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, u2 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 however which arises from the introduction of the new