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 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 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 t, of independent variables, each of which may be 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 s integrations are effected, then will the resulting value of V be a funtion of u and of the a unaccented variables to be determined. Bat after the introduction of u, the function V has the property of satisfying a partial differential equation of the second order, and by an application of the Calculus 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 a accented variables, the remaining one being a simple algebraic function whose form continues unchanged, whatever that of the first factor may be. * This function in its original form is given by where cUdy'dJ represents the volume of say element of the attracting body of which p' is the density and J, y', i' are the rectangular co-ordinates; x, y, r 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, r p'dddyW J {(j/ - *)• + (y' - y)«+ (• - r)» + e^T both integrals being supposed to extend over the whole volume of the attracting body. \ The chief object of the present paper is to resolve the problem in the more extended signification which we have endeavoured to explain 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 relations, the rational and entire factor in p still remains rational and entire of the same degree, and may under its altered form be expanded in a series of a finite number 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 a of the new variables entering into V, 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 constantly equal to zero so long as the attracted point p remains within the ellipsoid, but becoming equal to a determinate function of the co-ordinates of p, when /) is situated in the exterior space. Instead too of seeking directly the v&lae of V. all its differentials have first been deduced, and thence the value of V obtained by integration. This slight modification has been given to our method, both because it renders the determination of V in the case considered more easy, and may likewise be usefully employed in the more general one before mentioned. The other application is remarkable both on account i>f the simplicity of the results to which it leads, and of their analogy with those obtained by Laplace. (MSc. Oil. Liv. III. Chap. 2). In fact, it would be easy to shew that these last are only particular cases cf the mere general ones contained in the article now under notice. The general solution of the partial differential equation of the second order, deducible from the serenth and three following articles of this paper, and in which the principal variable V is a function of < +1 independent variables, is capable of being applied with advantage to various interesting physico-mathematical enquiries. Indeed the law of the distribution of heat in a body of ellipsoidal figure, and that of the motion of a nonelastic fluid over «> solid obstacle of similar form, may be thence almost immediately deduced; but the length of our paper entirely precludes any tbing more than an alius;'on to these applications on the present occasion. 1. The object of the present paper will be to exhibit certain general analytical formula;, from which may be deduced as a very particular case the values of the attractions exerted by ellipsoids upon any exterior or interior point, supposing their densities to be represented by functions of great generality. Let us therefore begin with considering o as a function of the 5 independent variables *i» **• xi a?»» and let us afterwards form the function V»( </*,'</<<&; d*;.p the sign / serving to indicate s integrations relative to the variables Xj', xt', x,',... x,\ and similar to the double and triple ones employed in the solution of geometrical and mechanical problems. Then it is easy to perceive that the function V will satisfy the partial differential equation seeing that in consequence of the denominator of the expression (1), every one of its elements satisfies for F"to the equation (2). To give an example of the manner in which the multiple integral is to be taken, we may conceive it to comprise all the real values both positive and r.egative of the variables as/as,', ...a,', which satisfy the condition »/., *En '**'•> • « / \ -hi + :\+-%+.... + —.t<l (c), °1 °» °i a« the symbol <, as is the case also in what follows, not excluding equality. 2. In order to avoid the difficulties usually attendant on integrations like those of the formula (1), it will here be convenient to notice two or three very simple properties of the function V. In the first place, then, it is clear that the denominator of the formula (1) may always be expanded in an ascending series of the entire powers of the increments of the variables as,, xt,...xn «, and their various products by means of Taylor's Theorem, unless we have simultaneously *,=»#,', xt = xt' x,*=x,' and « = 0; and therefore V may always be expanded in a series of like form, unless the e + l equations immediately preceding are all satisfied for one at least of the elements of V. It is thus evident that the function V possesses the property in question, except only when the two conditions 2i + aJ+5J + ... + 54<l and M = 0 (3) are satisfied simultaneously, considering as we shall in what |