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IX. On the Calculation of Attractions, and the Figure of the Earth. 
By C. J. Hargreave, B.A., of University College, London. Communicated by 
John T. Graves, Esq., A.M., F.R.S., of the Inner Temple. 
Received January 14, — Read February 11, 1841. 
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 accord- 
ing to any given law, would assume, when every particle is acted upon by the attrac- 
tion 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 so- 
lution 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 illus- 
trate 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 re- 
ferred 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 solu- 
tion 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 pub- 
lished his method, which has the great advantage of being more direct, though 
equally limited. 
The method given in the following paper does not appear to be confined in its 
operation to any particular class of spheroids ; since the coefficients of the series, into 
which the required function is developed, are determined absolutely, without any re- 
ference to the form of the spheroid to which they are about to be applied. The prin- 
cipal change consists in the different manner of treating this partial differential equa- 
tion. Laplace and the subsequent writers on this equation, both as applied to the 
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