same as that in No. 9 of LAGRANGE's memoir, becomes unsatisfactory and undeserving the name of proof, except when all the functions ) di(0) ddr(0) &c. are finite quantities at dy both the limits, and likewise for every intermediate value of y; which will not be the case unless u be a rational and integral function of μ', ✔— μ'. cos. ', √1-u. sin. '. Luckily, however, the author's own formulas suggest a clear and satisfactory way of determining this point without any transformation or the help of difficult integrations. LAGRANGE has proved in the most incontestible manner, that the theorem of LAPLACE cannot be true unless the following equation likewise take place, viz. ds dr =- 2πa.v: = s + a (as) and hence, it is plain, we shall be able to discover what function u is of the sines and cosines of the angles and, by considering in what manner these quantities enter into the equivalent expression on the left-hand side. Let x cos. 0 = μ‚ y=√1 — μ2. cos. π, z = √1—μ3. sin. ; x'= cos. 8' =μ', y' = √1— μ22. cos. w', z' = √1—μ3. sin. '; then* g=μμ2 + √π-μ3. √1—μ". cos. (≈'— π) = xx'+yy′+≈≈′; and, by substitution, we shall get 18 I // = √r2— 2ra. (xx' + yy'+ zz')+ therefore is a function of x, y, z; and ‡. † + a will likewise be a function of the same quantities: but * See No. 3 of my paper. ds Z; and because x, y, z are constant quantities, there will be the same powers and combinations of them in the integral as in the fluxion, the coefficients merely being changed: therefore the expression s+a (2) is likewise a function of x, y, and farther it is such a function as, being expanded into a series, can coincide only with a rational and integral function of the same quantities, consisting of a finite or infinite number of terms. Therefore the equation ds dr // s + a Ωπα . υ cannot take place unless u is a like function of x, y, z. ບ . The review which we have here taken of LAGRANGE'S memoir, and the observations we have made upon it, confirm the conclusions drawn in my paper, and throw additional light upon this difficult subject. We are indebted to the skill and abilities of LAPLACE for the invention of an equation in partial fluxions which has already contributed much to advance our knowledge of that branch of physical astronomy which relates to the figure of the planets, and which promises still greater improvements by suggesting new methods and removing the obstacles that have impeded the researches of former mathematicians: but he has not been so happy in founding his application of this invention on the theorem concerning the attractions at the surfaces of spheroids. It is impossible to deny that this theorem, as it is delivered in the Mécanique Céleste, is unsupported by any demonstrative proof; and that the extent of it has not been well understood. Instead of the indirect investigation which LAPLACE has followed, it were to be wished, for the sake of greater clearness and of avoiding the subtilties that occur in his analysis, that the attractions of such spheroids, as have been shown to fall under his method, were deduced directly from the function which expresses their radii: and on this account some degree of consideration may perhaps be attached to another paper of mine, presented to the Society about the middle of July last, in which an attempt is made to accomplish the object here mentioned. Oct. 30, 1811. II. On the Attractions of an extensive Class of Spheroids. By Communicated by Henry Brougham, Esq. J. Ivory, A. M. F. R. S. M. P. Read November 14, 1811. N In this discourse I propose to investigate the attractions of a very extensive class of spheroids, of which the general description is, that they have their radii expressed by rational and integral functions of three rectangular co-ordinates of a point in the surface of a sphere. Such spheroids may be characterized more precisely in the following manner: conceive a sphere of which the radius is unit, and three planes intersecting one another at right angles in the centre ; from any point in the surface of the sphere draw three perpendicular co-ordinates to the fixed planes, and through the same point in the surface likewise draw a right line from the centre, and cut off from that line a part equal to any rational and integral function of the three co-ordinates: then will the extremity of the part so cut off be a point in the surface of a spheroid of the kind alluded to; and all the points in the same surface will be determined by making the like construction for every point in the surface of the sphere. The term of a rational and integral function is not to be strictly confined here to such functions only as consist of a finite number of terms; it may include infinite serieses, provided they are converging ones; and it may even be extended to any algebraic expressions that can be expanded into such serieses. This class of spheroids comprehends the sphere, the ellipsoid, both sorts of elliptical spheroids of revolution, and an infinite number of other figures, as well such as can be described by the revolving of curves about their axes, as others which cannot be so generated. In the second chapter of the third book of the Mécanique Céleste, LAPLACE has treated of the attractions of spheroids of every kind; and in particular he has given a very ingenious method for computing the attractive forces of that class which in their figures approach nearly to spheres. In studying that work, I discovered that the learned author had fallen into an error in the proof of his fundamental theorem; in consequence of which he has represented his method as applicable to all spheroids whatever, provided they do not differ much from spheres; whereas in truth, when the error of calculation is corrected, and the demonstration made rigorous, his analysis is confined exclusively to that particular kind, described above, which it is proposed to make the subject of this discourse. I have already treated of this matter in a separate paper, in which I have pointed out the source of LAPLACE's mistake, and likewise have strictly demonstrated his method for the instances that properly fall within its scope. In farther considering the same subject, it occurred to me that the investigation in the second chapter of the third book of the Mécanique Céleste, however skilfully and ingeniously conceived, is nevertheless indirect, and is besides liable to another objection of |