A − ω² = C / (1 + ε²),
which gives
| ω² | = | 3 + ε² | tan−1ε − | 3 | . |
| 2πk²ρ | ε³ | ε² |
In the case of the earth, which is nearly spherical, we obtain by expanding the expression for ω² in powers of ε², rejecting the higher powers, and remarking that the ellipticity e = ½ε²,
ω² / 2πk²ρ = 4ε² / 15 = 8e / 15.
Now if m be the ratio of the centrifugal force to the intensity of gravity at the equator, and a = c(1 + e), then
m = aω² / 4⁄3πk²ρa, ∴ ω² / 2πk²ρ = 2⁄3m.
In the case of the earth it is a matter of observation that m = 1/289, hence the ellipticity
e = 5m / 4 = 1/231,
so that the ratio of the axes on the supposition of a homogeneous fluid earth is 230 : 231, as stated by Newton.