73 
extensive Class of Spheroids . 
solution is assumed as a necessary hypothesis without which 
the investigation will not succeed, in the other, is derived as 
a necessary consequence of a more general supposition. Here 
then the two methods are so much at variance, that if one be 
rigorous and exact, the other cannot be exculpated from the 
charge of erroneous or insufficient reasoning. This contra- 
diction between the preceding analysis and the procedure of 
Laplace is entirely consonant to tire conclusions obtained in 
my former paper alluded to in the beginning cf this discourse; 
and the origin of it is to be sought for in the error I there 
pointed out in the investigation of that geometer. It cannot 
be denied that an error of calculation does exist in the de- 
monstration of the theorem on which that author’s method is 
grounded: his reasoning is therefore imperfect and inconclu- 
sive; and the inferences he has drawn from it cannot be sup- 
ported in opposition to a rigorous analysis. 
10. The same procedure which has been applied to approxi- 
mate to the attractions of spheroids differing little from spheres, 
may likewise be employed to find accurate expressions in 
serieses of the attractive forces of any spheroid, provided the 
radius of it be such a function as the analysis requires. In 
both cases the research turns upon the same sort of integrals. 
Resume the general term of the series for the attractive force 
on a point without the surface, viz. 
st arbitraire, mais die derive du devdoppement en serie, des attractions des sphe~ 
■“ roides.” 
In this formula it is necessarily implied, that y is-a rational and integral function 
u'(°l u(0 
cf three rectangular co-ordinates of a sphere ; because all the terms — , — &c. 
a tr 
are necessarily such functions. 
MDCCCXII. L 
