the Attractions of Spheroids of every Description. $ 
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 Mecanique Celeste , 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 par- 
ticular 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 sur- 
face ; 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 qucesitum 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 
