7 
the Attractions of Spheroids of every Description . 
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 develope- 
ment 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 con- 
sequences which flow from it, than it is curious in an analy- 
tical 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 Mecanique Ce- 
leste, the proposition just mentioned is enunciated in the most 
general manner, comprehending every case in which the at- 
tractive 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 pf- 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) f lb. Equat. (2) 
