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Ill. On the Theory of the Figure of the Planets contained in 
the Third Book of the Mécanique Céleste. By J. Ivory; 
Esq. M.A. F.RS. 
(Continued from vol. Ixvi. p. 439.] 
GOME apology may perhaps be due from me to the readers 
of the Philosophical Magazine, for drawing their attention 
to a subject so much neglected in the present times as that 
which I have undertaken to discuss. It seems to be the ge- 
neral disposition to rest entirely contented with what has al- 
ready been accomplished in the theory of the figure of the 
planets. - But as all the useful and most important results had 
already been obtained by Clairaut, we ought, in order to be 
consistent, to go back to the luminous and elegant theory of 
that excellent geometer. It will be answered, that the theory 
in question is imperfect, inasmuch as it merely demonstrates 
the equilibrium of a planet when it is supposed to have the 
figure of an oblate spheroid of revolution. The objection is 
of great weight; and it never can be admitted that the suc- 
cessors of Newton have perfected his system, until the figure 
of a planet is clearly deduced from the laws of equilibrium 
without any adventitious supposition. The learned researches 
of Legendre and Laplace have generally been supposed to 
obviate the foregoing objection, at least when the bodies are 
nearly spherical, as is universally true of the planets. But an 
attentive examination will show that there is still something 
imperfect in the theory of the illustrious geometers we have 
named. ‘There is, in fact, involved in it a hidden principle 
which is equivalent to the gratuitous supposition of Clairaut. 
The perpendicularity of gravity to the outer surface is common 
to both; but, as this principle is alone insufficient, Clairaut 
assumes the figure of an oblate spheroid, while the analytical 
method employed by Legendre and Laplace dispenses with any 
such supposition in the particular case they have considered. 
But is it possible that the varying of an abstract method of 
calculation can in any respect alter the physical foundations 
of the problem? In order to solve this difficulty, it is to be 
observed that Legendre and Laplace proceed upon a deficient 
theory of equilibrium; a necessary condition is omitted; but 
it so happens that, in the particular circumstances of the pro- 
blem to which they have confined their attention, the omission 
may be made without leading to error in a first approximation, 
and in a first approximation only. This sufficiently explains 
why a result is obtained that agrees with the solution of Clai- 
raut. But if the result of a first approximation be correct, it 
is 
