3.1 Preliminaries

First, let's write down formulas for quantities that we will need lateron (Heiskanen and Moritz, 1967). First the relationship between Earth flattening parameter $f$, semi-major and semi-minor axes $a$ and $b$:

$$f =\frac{a-b}{a}$$ (1)

$$b =a(1-f)$$ (2)

$$E =\sqrt{a^{2}-b^{2}}$$ (linear eccentricity) (3)

$$e =\frac{E}{a}$$ (first numerical eccentricity) (4)

$$e' =\frac{E}{b}$$ (second numerical eccentricity) (5)

$$m =\frac{\omega^{2}a^{2}b}{GM}$$ (an abbreviation) (6)

$$\gamma_{a} =\frac{GM}{ab}\left(1-\frac{3}{2}m-\frac{3}{14}e'm\right)$$ (gravity acc. at equator) (7)

$$\gamma_{b} =\frac{GM}{a^{2}}\left(1+m+\frac{3}{7}e'm\right)$$ (gravity acc. at poles) (8)

$$x =\frac{a\cos\varphi\cos\lambda}{\sqrt{1-e^{2}\sin^{2}\varphi}}$$ (Cartesian coordinates (9)

$$y =\frac{a\cos\varphi\sin\lambda}{\sqrt{1-e^{2}\sin^{2}\varphi}}$$ relative to center (10)

$$z =\frac{a(1-e^{2})\sin\varphi}{\sqrt{1-e^{2}\sin^{2}\varphi}}$$ of Earth) (11)