II 
Mr. Ivory on the Attractions of an 
V = k ( ^ -j- 6^ . r -j- b^ . r* -{- b ^ . r 3 &c- 5 
then by expanding the radical in the formula (4) into a series 
of a similar form, and equating the corresponding terms, we 
shall get 
6(0 =/// 
rfR' . dy! . dm' . q(0 
* 
R* 
— i 
In this value of b^\ the integration with regard to dR' can- 
not be executed from R/ = 0, as in the former case ; because 
this expansion of V necessarily supposes that the attracted 
point is included within all the attracting matter: let R be 
what R' becomes at the surface of the spheroid, which is the 
outer surface bounding the attracting matter, and let p be the 
radius of the inner surface ; then, with respect to the matter 
between the two surfaces, and for a point within them both, 
we shall have 
fc(i) = SJ llh - jhl • V- *-'• tf 0 - (6). 
In the case of i — 2, the expression of the coefficient takes 
a particular form : for 
and, by integrating, 
6 (2> —Jf { log. R' - log. f } . dy' . dx ' . Q (2) . 
Let us now seek an expression of the force with which the 
whole spheroid attracts a point within the surface. For this 
purpose we shall suppose p to denote the radius of a sphere 
which completely envelops the spheroid : and we shall deter- 
mine ; first, the value of V, relatively to the matter between 
the spheroid and the sphere ; secondly, its value, relatively to 
* Mec. Cel. Liv. je, No. 15. 
