Relation between the reduced and geographical latitudes

We have derived Equation (2.164), which shows how the field varies with the reduced latitude p on the surface of the spheroid. The reduced latitude is the angle between the radius vector and the equatorial plane. Fig. 2.7c. Also, it is useful to study the function y — y q ), where tp is the geographical latitude. This angle is formed by the normal to the ellipsoid at the given point p and the equatorial plane. Fig. 2.7b. First, we find expressions for coordinates v, y of the meridian ellipse. Its equation is 

The value of the derivative is equal to the tangent of the angle, formed by the tangent with the x-axis  

This formula was introduced by the Italian geodesist Somigliana in 1929 and defines the behavior of the gravitational field on the level surface of the spheroid for any distribution of density inside, as long as the outer surface remains equipotential. 

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