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XX. On the Attractions of homogeneous Ellipsoids. By James 
Ivory, A. M. Communicated by Henry Brougham, Esq. 
F. R. S. 
Read June 15, 1809. 
1. The theory of the figures of the planets involves in it 
two distinct researches. In the first of these, it is required to 
determine the force with which a body, of a given figure and 
density, would attract a particle of matter, occupying any pro- 
posed situation : in the second, the subject of investigation is 
the figure itself, which a mass of matter, wholly or partly 
fluid, would assume, by the joint effect of the mutual attrac- 
tion of its particles, and a centrifugal force arising from a ro- 
tatory motion about an axis. To render the second of these 
inquiries more exactly conformable to what actually takes 
place in nature, the influence of the attractions of the several 
bodies, that compose the planetary system, ought to be super- 
added to the forces already mentioned. 
It is the first of these two researches, of which we propose 
to treat at present ; and we shall even confine our attention 
to homogeneous bodies, bounded by finite surfaces of the se- 
cond order. 
The theory of the attractions of spherical bodies is delivered 
by Sir Isaac Newton in the first book of the Principia.* In 
the same place the illustrious author lays down a method for 
* Sect. 12. 
