49 
extensive Class of Spheroids. 
to lay down the analytical propositions on which the investi' 
gation is founded : the second section will contain the solution 
of the problem under consideration. 
One more preliminary observation it is proper to add. The 
problem of attractions contains two cases ; when the density 
of the attracting body is uniform throughout ; when it varies 
according to any given law ; it is in the first of these two 
cases that the chief difficulties occur ; and as I have nothing 
new to add on the second case, I shall here confine my atten- 
tion to homogeneous spheroids, unit being supposed to denote 
the density. 
I. 
Preliminary Investigations. 
1. Let {jo denote the cosine of an angle, and let 
/= — p; 
then the truth of the following equation in partial Auctions 
will be proved merely by performing the operations indicated, 
viz. 
) =o 
Now put S = ■ ■ 1 , ■ ; then 
r fzn + i ’ 
(e) = -(“ + 0-£=&| 
(|) = 2B + 1 -fm j 
therefore, on account of the first equation, we shall obtain by 
substitution. 
MDCCCCXII. 
H 
