the Attractions of Spheroids of every Description. 31 
now the exact coincidence of these two equations is proved by 
observing that the series into which v is multiplied is equal to 
unit ; for it is equal to 1 — V 1 — « x when x is put equal to 
unit. 
7. I have now explained at sufficient length my objections 
to Laplace's demonstration, and the reasons on which they 
are founded. The subject is abstruse and subtile ; on which 
account I have taken all the pains I could to make the pro- 
cesses as clear as the nature of such a discussion would per- 
mit ; and I have endeavoured to confirm the conclusions I 
wished to establish by investigating them in more ways than 
one. It appears, from what has been shown, that Laplace’s 
theorem, which in the law of attraction that takes place 
in nature is contained in Equation (2), No. 10, Liv. 30. of 
the Mecanique Celeste , is neither true of all spheroids that 
nearly approach the spherical figure as the author thought, 
nor is it strictly demonstrated in any case. It is exclusively 
confined to that class of spheroids which, while they differ 
little from spheres, likewise have their radii expressed by 
rational and integral functions of a point in the surface of a 
sphere : in this hypothesis Laplace’s equation has been rigor- 
ously demonstrated in the preceding pages ; and it is to such 
spheroids only that his ingenious method, which is founded 
on that equation, can be applied. 
And here a question occurs. Since the solution of the pro- 
blem of attractions contained in the Mecanique Celeste is not a 
universal method for all spheroids differing little from spheres, 
as the author conceived it to be, but is really limited to one 
particular class of spheroids ; it may be asked, how far will 
this limitation affect the physical theories he has built on his 
