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IX. On the calculation of attractions, and the figure of the earth

1841; Royal Society; Volume: 131; Linguagem: Inglês

10.1098/rstl.1841.0011

2053-9223

Charles James Hargreave,

Planetary Science and Exploration

The principal object of the calculations contained in the following paper, is to investigate the figure which a fluid, consisting of portions varying in density according to any given law, would assume, when every particle is acted upon by the attraction of every other and by a centrifugal force arising from rotatory motion. To what extent this may have been the original condition of the earth, is a doubtful question; and although observation does not fully warrant this supposition of the regular arrangement of parts, it has necessarily been made the foundation of most of the mathematical calculations connected with the investigation. Before proceeding to this problem, it is necessary to calculate the attraction of a body of any given figure, and consisting of strata, varying in their densities according to any given law; and it is in this problem that the principal difficulty lies. The elegant method of solution discovered by Laplace is well known; and I have followed his steps as far as the point where the equation, known by his name, first appears. In order to illustrate the nature of the deviation which I have there made, it will be necessary to mention some of the principal steps of the two methods. By means of a theorem, which Laplace laid down as true of all spheroids that differ but little from spheres, and the properties of the integral of the equation referred to, he was enabled to substitute the easy rules of differentiation for the more complicated inverse processes, and thus to compute the attraction of that class of figures. It has, however, been since discovered by Mr. Ivory, that this theorem is true only of spheroids of a particular kind; and, consequently, to this kind the solution of the problem is restricted. This defect, and the indirectness of his analysis, led other mathematicians to consider the question; and, in 1811, Mr. Ivory published his method, which has the great advantage of being more direct, though equally limited.