**7. Clairaut theorem**

At 1743 the French mathematical scientist Clairaut found a mathematical expression to represent the relation between the gravity measurements on the earth surface and the shape of the earth.

Supposing that.

ɑ = maximum radius of oblate spheroid.

b = minimum radius of oblate spheroid.

a and b are the major semi axes of an oblate spheroid of revolution. and write

$$\mathbf{b} = \mathbf{a} \ (\mathbf{1} - \mathbf{f}).\tag{27}$$

$$\mathbf{f} = (\mathbf{a} \mathbf{-} \mathbf{b}) / \mathbf{a} \tag{28}$$

F = oblateness or ellipticity of the spheroid for spherical body

$$\left( \left( \mathbf{x}^2 + \mathbf{y}^2 \right) / \mathbf{a}^2 \right) + \left( \mathbf{z}^2 / \left( \mathbf{a}^2 \left( \mathbf{1} \mathbf{-2} \, \mathbf{f} \right) \right) \right) = \mathbf{1} \tag{29}$$

That is

$$\mathbf{x}^2 + \mathbf{y}^2 + \mathbf{z}^2 \ (\mathbf{1} + 2\mathbf{f}) = \mathbf{a}^2 \tag{30}$$

if is the colatitude angle *Ө* the above equation written as

$$\mathbf{r}^2 \left( \mathbf{1} + \mathbf{2} \, \mathbf{f} \, \mathbf{C} \mathbf{s}^2 \, \boldsymbol{\Theta} \right) = \mathbf{a}^2 \tag{31}$$

or

$$\mathbf{a}^2/\mathbf{r}^2 = \left(\mathbf{1} + 2\mathbf{f}\,\mathbf{C}\mathbf{s}^2\Theta\right) \tag{32}$$

Considering the earth as an oblate spheroid, rotate around its axis and the variation in gravity measurements on the earth, Clairaut obtained the following theorem:-

$$\mathbf{g} = \mathbf{g}\_{\epsilon} \left[ \mathbf{1} + \left( \frac{5}{2} \mathbf{C} - f \right) \sin^{2} \Phi \right] \tag{33}$$

Where

g = Theoretical gravity value at any point on the earth according to its latitude angle.

gₑ = gravity at the equator.

Ф = latitude angle

$$\mathbf{C} = \mathbf{o}^2 \mathbf{a} / \mathbf{g}\_{\varepsilon} = (\mathbf{C} \text{centrifugal force} / \text{Attraction force}) \text{ at equator} \tag{34}$$

(where latitude angle equal zero) The variable of sin<sup>2</sup> Ф which is (5c/2) – *f* Represent a gravitational flattening (*β*)

Which is written in other expression as following:

$$
\beta = \frac{\mathbf{g}\_p - \mathbf{g}\_e}{\mathbf{g}\_e} \tag{35}
$$

gₑ = Gravity at equator gₚ = Gravity at pole The Clairaut's theorem also written in other expression:

$$\mathbf{g}\_{\circ} = \mathbf{g}\_{\circ} \left( \mathbf{1} + \beta \, \sin^{2} \Phi \right) \tag{36}$$

The value of the factor *β* can be obtained from measuring a lot of absolute gravity values at different locations at the earth surface. The slope of the best fit line of the gravity value versus the latitude angle represent the factor *β* multiplied by the gravity at the equator gₑ.

$$\text{Considering}\\
f = \mathbf{1}/2\mathbf{97} \tag{37}$$

The determination of gravity values at the sea level over the world led to obtained the following equation:

$$\mathbf{G} = \mathfrak{Y}\mathbf{8}.049 \left( 1 + 0.0052884 \sin^2 \Phi - 0.0000059 \sin^2 2\Phi \right) \tag{38}$$

Grand and West, [2].

*Gravity Field Theory DOI: http://dx.doi.org/10.5772/intechopen.99959*

This equation called the international gravity formula, or the 1939's equation, where the gravity value at equator (gₑ) is (978.049 gal), which calculated statistically from measurements on the earth surface. The value of the factor of (sin2 Ф) represent the flattening factor effect, while the value of the factor of (sin2 2Ф) represent a correction value to fit the earth shape with rotated spheroid body shape. These two factors depends on the shape of the earth and speed of rotation of the earth.

Depending on the principles of gravity anomalies the earth seems as triaxle ellipsoid. The long axis of the earth lying 10° west of Greenwich and the difference between radiuses of the earth may be within 150 � 58 meters, which is considered very low variation relative to the average radius of the earth therefore it is neglected in most cases.

There is another formulas such as Helmert, 1901 formula

$$\mathbf{g}\_o = \text{978.030 } \left( \mathbf{1} + 0.005302 \sin^2 \Phi - 0.000007 \sin^2 2\Phi \right) \tag{39}$$

the radius of the according to this equation are: a = 6378200 m. b = 6356818 m and

$$f = \mathbf{1}/298.2\tag{40}$$

This formula used in old gravity measurements.

Other formula used in 1917 in the United States of America using the following formula:

$$\mathbf{g}\_{\bullet} = \text{978.039 } \left( \mathbf{1} + \text{0.005294 } \sin^{2} \Phi - \text{0.000007 } \sin^{2} 2\Phi \right) \tag{41}$$

The above formula is obtained depending on 216 gravity stations in USA, 42 in Canada, 17 in Europe, and 73 gravity stations in India. The flattening value of this formula (1/297.4).

The international gravity formula 1967 which is adopted by the Geodetic Reference System (GRS-1967) with different factors values according to new observations [3]

$$\mathbf{g}\_{\circ} = \text{978.03185 } \left( \mathbf{1} + \text{0.0053024 } \sin^{2} \Phi \text{-0.0000059 } \sin^{2} 2\Phi \right) \tag{42}$$

This formula used to remove the variation in gravity with latitude (latitude correction). This formulae consider the earth as rotating ellipsoid without geologic or topographic complexities.

The GRS 1980 formula is

$$\mathbf{g}\_{\circ} = \text{978.0327} \left( \mathbf{1} + \text{0.0053024 } \sin^{2} \Phi \text{--} \mathbf{0}.0000058 \, \sin^{2} \Phi \right) \tag{43}$$

The gravity survey for small sites for examples in case of engineering studies the latitude correction approximated using the a correction factor 0.813 sin 2Ф mgal/ km in the north- south direction.
