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of different fluids of different densities, will be in equilibrium, and will for ever preserve its figure when it has the form of an elliptical spheroid of revolution oblate at the poles. It has likewise been proved that the same form is the only one capable of fulfilling the required conditions; which completes the solution of the problem in so far as it regards a mass entirely fluid.
The hypothesis of NEWTON, although most judicious, and best adapted for simplifying the investigation, is nevertheless quite arbitrary, and indeed does not seem to agree well with what is observed at the surface of the earth. Had the terrestrial globe been once entirely fluid, the heterogeneous matters of which it consists, must have taken an arrangement depending on their densities; the substances of greatest density would ultimately have settled at the centre, and those of least density at the surface; and in proceeding from the centre to the surface, the changes of density would not have been very sudden, but slow and gradual and hardly perceptible for considerable depths. Admitting this hypothesis we should therefore expect to find all the matter at the earth's surface, or near it, little different in respect of density; which is quite contrary to experience, since nothing can be more unequal and irregular than the density of the substances that compose the upper strata of the earth. Many other phenomena are also inconsistent with that uniform arrangement of parts which seems to be a necessary consequence of the supposition that the earth was originally fluid: of this description are, the great elevation of the continents above the surface of the sea; the depth of the immense channels which contain the waters diffused over the surface of the globe; and the irregular
disposition of the land and water on the same surface. Besides all this, after a long discussion, in which every circumstance that can affect the question, has been duly weighed, it seems now to be ascertained, that the elliptical figure of the earth, cannot be reconciled with the actual measurements which have been made for the express purpose of bringing the theory to the test of experiment. The hypothesis of NEWTON is therefore not exactly consonant to observation: and we must infer that the solid part of the earth is not, at least in the present state of the globe, possessed of that regularity of figure, nor of that peculiar disposition of the internal strata, which would arise from the earth's having been originally fluid. Hence it becomes necessary to consider the question of the figure of the planets in a more enlarged point of view; to free it from all arbitrary suppositions, and to attempt such a solution of the problem, as shall apply to whatever figure or hypothesis may appear most agreeable to observation. It is in this way only, that theory and observation can mutually assist one another, and ultimately lead us to the truth-that theory can prompt observation, and observation perfect and confirm theory.
The celebrated French mathematician, D'ALEMBERT, was the first who contemplated the question of the figure of the planets in a general manner, by extending his researches to other figures than the elliptical spheroid. The difficulty is to investigate the attractive force of a body of any proposed figure, and composed of strata that vary in their densities, according to any given law. D'ALEMBERT invented a method for this purpose which, although it is very ingenious, and so general as to apply in a great variety of cases, is nevertheless
destitute of that simplicity which is absolutely necessary for advancing our knowledge in an enquiry so complicated in all respects.
LAPLACE, to whom every part of physical astronomy owes so much, has been very successful in improving that branch of it which relates to the figure of the planets, and to other questions with which this is connected. The foundation of his researches on this subject, is laid in the second chapter of the third book of the Mécanique Céleste, where he treats of the attractions of spheroids in general, and more particularly of such as differ but little from spheres. The investigation required in this part of physics, if it be guided by the desire of obtaining useful conclusions, is not only extremely difficult, but of a nature so nice and delicate, as would at first seem to elude the ordinary methods of analysis, and to require particular contrivances adapted to the exigencies of the case. When a fluid covering a solid body, has assumed a permanent figure, that figure will depend upon the gravity at the surface; while the same gravity, being the combined effect of the attractions of all the molecules of the compound body, is itself produced by the form of the surface. Thus the figure of the surface is in a manner both a datum and quæsitum of the problem; and the skill of the analyst must be directed to find an expression of the intensity of the attractive force which shall be sufficiently simple, and shall likewise preserve in it the elements of the figure of the attracting solid. All these conditions are fulfilled in the skilful solution of the problem of attractions given by LAPLACE, in which the relation between the radius of the spheroid and the series for the attractive force on a point without, or within, the surface, or on it, is
deduced in a manner admirably simple, when the complicated nature of the question is considered.
In order to give a succinct view of the plan of analysis pursued by LAPLACE, we must begin with observing that he does not seek directly an expression of the attractive force, but that he investigates the value of another function from which the attractive force in any proposed direction, may be derived by easy algebraic operations. This function, which in the law of attraction that obtains in nature, is the sum of all the molecules of the attracting solid, divided by their respective distances from the attracted point, he expands in all cases into a series, containing the descending powers of the distance of the attracted point from the center, when that point is without the surface; but the ascending powers of the same distance, when the attracted point is within the surface: and the question is, to determine the coefficients of the several terms of the expansion. In the first place, it is proved that every one of the coefficients satisfies an equation in partial fluxions, first noticed by the author himself, and from the skilful use of which, all the advantages peculiar to his method are derived. LAPLACE next lays down a theorem, which, he affirms, is true at the surfaces of all spheroids that differ but little from spheres; hence he deduces the value of an expression, which is the sum of all the coefficients sought respectively multiplied by a known number; and, what is remarkable, the value alluded to, is found to be proportional to the difference between that radius of the spheroid which is drawn through the attracted point, and the radius of the sphere nearly equal to the spheroid. The circumstances we have now mentioned, suggest an elegant solution of the problem, and one that has
the advantage of expressing the radius of the spheroid and the series for the attractive force, by means of the same functions. For in order to find the coefficients sought, we have only to develope the difference between the radius of the spheroid, and the radius of the sphere, into a series of parts, every one of which shall satisfy the equation in partial fluxions and LAPLACE not only gives a method for computing the several parts, but he likewise proves that the developement is unique, or can be made no more ways than one.
The solution, of which we have endeavoured to give a concise notion, is not more important for the physical consequences which flow from it, than it is curious in an analytical point of view, for the singular art with which the author has avoided the complicated integrations that naturally occur in the investigation, and has substituted in their room the easy operations of the direct method of fluxions. He has been enabled to do this by the help of the theorem which he had discovered to be true at the surfaces of all spheroids that nearly approach the spherical figure. In the Mécanique Cé-leste, the proposition just mentioned is enunciated in the mostgeneral manner, comprehending every case in which the attractive force is proportional to any power of the distance be tween the attracting particles :* but in order to avoid every discussion not essential to the main scope of this discourse, I shall chiefly confine my attention to the case of nature in which the attraction follows the inverse proportion of the square of the distance; this being the only case which it is really interesting to consider, because it is the only one that enters into the inquiry concerning the figure of the planets. The Liv. 3. No. 10. Equat, (1). + Ib. Equat. (2)