On the Attractions of an extensive Class of Spheroids. 47 
include infinite serieses, provided they are converging ones ; 
and it may even be extended to any algebraic expressions that 
can be expanded into such serieses. This class of spheroids 
comprehends the sphere, the ellipsoid, both sorts of ellip- 
tical spheroids of revolution, and an infinite number of other 
figures, as well such as can be described by the revolving 
of curves about their axes, as others which cannot be so- 
generated. 
In the second chapter of the third book of the Mecanique 
Celeste , Laplace has treated of the attractions of spheroids of 
every kind ; and in particular he has given a very ingenious 
method for computing the attractive forces of that class which 
in their figures approach nearly to spheres. In studying that 
work, I discovered that the learned author had fallen into an 
error in the proof of his fundamental theorem ; in consequence 
of which he has represented his method as applicable to all 
spheroids whatever, provided they do not differ much from 
spheres ; whereas in truth, when the error of calculation is 
corrected, and the demonstration made rigorous, his analysis 
is confined exclusively to that particular kind, described above, 
which it is proposed to make the subject of this discourse. I 
have already treated of this matter in a separate paper, in 
which I have pointed out the source of Laplace's mistake, 
and likewise have strictly demonstrated his method for the 
instances that properly fail within its scope. In farther con- 
sidering the same subject, it occurred to me that the investi- 
gation in the second chapter of the third book of the Mecanique 
Celeste , however skilfully and ingeniously conceived, is never- 
theless indirect, and is besides liable to another objection of 
