When ρ is expressed in terms of c, this equation can be integrated. We infer then that a rotating spheroid of very small ellipticity, composed of fluid homogeneous strata such as we have specified, will be in equilibrium; and when the law of the density is expressed, the law of the corresponding ellipticities will follow.
If we put M for the mass of the spheroid, then
| M = | 4π | ∫c0 ρd{c³(1 + 2e)}; and m = | c³ | · | 4π² | , |
| 3 | M | t² |
and putting c = c0 in the equation expressing the condition of equilibrium, we find
| M(2e − m) = | 4 | π · | 6 | ∫c0 ρ d(ec5). |
| 3 | 5c² |
Making these substitutions in the expressions for the forces at the surface, and putting r/c = 1 + e − e(h/c)², we get
| G cos φ = | Mk² | {1 − e − | 3 | m + ( | 5 | m − 2e) | h² | } | f |
| ac | 2 | 2 | c² | c |
| G sin φ = | Mk² | {1 + e − | 3 | m + ( | 5 | m − 2e) | h² | } | h | . |
| ac | 2 | 2 | c² | c |
Here G is gravity in the latitude φ, and a the radius of the equator. Since
sec φ = (c/f){1 + e + (eh²/c²)},