Attractions of Spheroids of. every Description. 35 
tion V = Gained m No. 5 of the preceding paper 
will become 
V = ±£ + *: 
3 r 1 
which is equivalent to the equation in the first paragraph of 
No. 2 of Lagrange’s memoir, the only difference being in 
the characters employed. And if we treat this equation as in 
No. 6 of my paper, and suppose that the term multiplied by 
r — a vanishes when r = a, we shall get 
±VJ r a 
27 t . a ' 
3 
i s + a [%) — 0 
when the attracted point is in the surface of the spheroid : and 
these equations are the very same with those which Lagrange 
has investigated, by a process entirely similar, in the remain- 
ing part of No. 2. 
The equation ■§• s -f- a (^) = 0, is considered by Lagrange 
in No. 3. As this equation was obtained by reasonings which 
are independent on the nature of the molecules in the differ- 
ence between the spheroid and the sphere, it ought to be true 
for all values of the function t/ which expresses the thickness 
of those molecules. In order to examine this point, Lagrange 
supposes 1/ to be a constant quantity ; and on this supposition he 
finds that in fact the equation f- s + a J =0, does not take 
place, but that the true equation is \ s -j- a (E) = - a’. «. 
Here then there is certainly a great difficulty : for the very 
same reasonings which prove \ s -{- a i~\ = 0, on the sup- 
position of Laplace that the sphere touches the spheroid at 
F 2 
