24 Mr. Ivory on the Method of computing 
no manner of connection with any thing touched upon in the 
whole course of his demonstration. In the rigorous investiga- 
tion, the rules of the integral calculus are necessary; whereas 
the reasoning of Laplace requires only the direct method of 
fluxions. Besides his proof goes too far ; for it applies to all 
spheroids that approach nearly to the spherical figure : but 
the method, when it is strictly analyzed, is limited to those 
spheroids of the same description which have their radii ex- 
pressed by rational and integral functions of three rectangular 
co-ordinates of a point in the surface of a sphere. We may 
even infer from what Laplace himself has proved that his 
method is confined exclusively to such spheroids : for he has 
shewn that the expression fory is not arbitrary, but that it 
depends upon the series for V ;* whence it follows that it can 
only be such a function as is mentioned above, and as we have 
supposed it to be. 
5. In order still farther to confirm the conclusions already 
obtained I shall now show that Laplace’s method for the 
attractions of spheroids that differ but little from spheres is 
contained in the formula (E) from which it may be deduced 
without the intervention of his theorem relating to the attrac- 
tion at the surface. 
Conceive a spheroid whose radius is p = . (1 -f- » . y) as 
before ; and also a sphere, whose radius is a, concentric with 
the spheroid ; and let r denote the distance of an attracted point 
situate in the prolongation of p, from the common centre: 
then the value of V relatively to the sphere will be = ~ 
and if dm denote one of the molecules of the excess of the 
* Liv. 3, No. 11. 
