XII. On the Figure assumed by a Flud Homogeneous 
Mass, whose Particles are acted on by their mutual 
Attraction, and by small extraneous Forces. 
By G. B. AIRY, B. A. 
OF TRINITY COLLEGE, 
AND FELLOW OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY, 
{Read March 15, 1824.] 
Tue principal difficulty in the solution of this problem, 
consists in the investigation of the attraction of any spheroid 
(differing little from a sphere) upon a point in its surface. This 
has been found by Laplace, in a manner so general, and by an 
analysis so powerful, that any new investigations might seem 
entirely unnecessary. But the abstruse nature of that analysis, 
it must be acknowledged, is such as to make a more simple in- 
vestigation desirable: and the obscurities which have led La- 
place himself into error, serve to shew the value of a process 
which involves nothing more difficult than the common appli- 
cations of the differential calculus. I venture to indulge in a hope 
that the solution which I have the honor to lay before this Society, 
imperfect as it may be, will tend to make this subject more 
accessible to those who have hitherto been deterred from pur- 
suing it by the mass of analysis in the Mecanique Celeste. 
With the exception of the proposition which reduces the 
discovery of the attraction in any direction to the investigation 
of the value of a single integral, the process pursued in_ this 
Paper is entirely different from that of Laplace. The theory 
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