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ON THE DETERMINATION
EXTERIOR AND INTERIOR
ATTRACTIONS OF ELLIPSOIDS
From the Transactions of the Cambridge Philosophical Socitty, 1835. [Read May 6, 1833.]
ON THE DETERMINATION OF THE EXTERIOR AND INTERIOR ATTRACTIONS OF ELLIPSOIDS OF VARIABLE DENSITIES.
Thk determination of the attractions of ellipsoids, even on the hypothesis of a uniform density, has, on account of the utility and difficulty of the prohlem, 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, M1 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 variable u, depends on the property which a certain function V* then possesses of satisfying a partial differential equation, whenever the law of the attraction is inversely as any power n of the distance. For by a proper application of this equation we may avoid all the difficulty usually presented by the integrations, and at the same time find the required attractions when the density p is expressed by the product of two factors, one of which is a simple algebraic quantity, and the remaining one any rational and entire function of the rectangular co-ordinates of the element to which p belongs.
The original problem being thus brought completely within the pale of analysis, is no longer confined as it were to the three dimensions of space. In fact, p may represent a function of any number », of independent variables, each of which may he marked with an accent, in order to distinguish this first system from another system of s analogous and unaccented variables, to be afterwards noticed, and V may represent the value of a multiple integral of s dimensions, of which every element is expressed by a fraction having for numerator the continued product of p into. the elements of all the accented variables, and for denominator a quantity containing the whole of these, with the unaccented ones also formed exactly on the model of the corresponding one in the value of V belonging to the original problem. Supposing now the auxiliary variable u is introduced, and the a integrations are effected, then will the resulting value of Fbe a funtion of u and of the s unaccented variables to
* This function in its original form is given by
F_ r p'dx'dy'dz'
- x)* + & - yf + (j/ - *)»} T'
where dx'dy'd/ represents the volume of any element of the attracting body of which ft is the density and a/, j/, z' are the rectangular co-ordinates; x, y, z being the co-ordinates of the attracted point p. But when we introduce the auxiliary variable u which is to be made equal to zero in the final result,
__ f p'dx'dy'dz'
both integrals being supposed to extend over the whole volume of the attracting body.