contained in the Third Book of the Mécanique Céleste. 87 
without which the equilibrium cannot subsist ; the other ascer- 
- tains the proportion of the forces necessary to make that figure 
possible. j 
We can now likewise discover, @ priort, why Laplace’s 
method, in the circumstances supposed, brings out a result 
which is exact as a first approximation. It arises from this, — 
that, when the fluid is a perfect sphere, the two conditions of 
equilibrium coincide in one; for Newton has proved that a 
spherical shell of homogeneous matter attracts a particle in 
the inside equally in opposite directions. What is exactly 
true in the case of a sphere, must be nearly so when the fluid 
approaches indefinitely to that figure; and in these circum- 
stances, the second condition being implied to a certain extent 
in the first, we obtain an approximation by means of the lat- 
ter alone. This, however, is true only of the first step in ap- 
proximating to the figure of a planet ; in proceeding further it 
becomes necessary to take in all the conditions of equilibrium, _ 
I shall only add one more remark on the analysis of La- 
place. The radius of the spheroid, drawn from the centre of 
gravity, is represented by a series of this form, viz. 
r=a-+aa SY?) + YO 4+ YO) + &c.}, 
the terms on the right-hand side being the development of the 
function y. This expression has nothing to do with the con- 
ditions of equilibrium; it belongs to every spheroid nearly 
spherical. Now the second of the conditions of the problem, 
or that one which determines the species of the figure neces- 
sary to the equilibrium, proves immediately that the radius 
can consist only of two terms, viz. 
r=a+aaY), 
In the method of Laplace the superfluous terms are got rid 
of by a particular mode of reasoning, which does not cohere 
well with the rest of the investigation, and certainly hurts the 
unity of solution. The process, indeed, is not long in the case 
of a uniform density; but when the fluid is composed of 
strata of variable density, the reasoning is both long and com- 
plicated, and the result is at length brought out by dint of 
calculation *. There is therefore a great advantage in solving 
the problem, as it ought to be solved, by the true principles 
of the case, both because it is more satisfactory to the mind, 
and because it is more simple. The observation here made is 
‘of the greater importance as it extends to the theory of the 
tides, and to the other questions treated of in the Mécanique 
Céleste, where the figure of the planets is concerned, the same 
form of the radius being assumed in all these cases. 
* Livre iii. § 29 & 30. 
In 
