Relationship between density, angular velocity, and spheroid eccentricity

Relationship between density, angular velocity, and spheroid eccentricity 

We will narrow our task and attempt to ehoose a funetion S sueh that its partial derivatives obey Equation (2.338). Of eourse, this does not allow us to find all possible figures of equilibrium, but for our purposes it is not important, because we are only interested to show that a rotating homogeneous spheroid, describing Earth, can be under certain conditions a figure of equilibrium. This fundamental fact was established by MacLauren in the 18th century. In this case we have 

In the previous section we found components of the attraction field due to masses of a homogeneous spheroid. Equations (2.326 and 2.327). Correspondingly, the components of the gravitational field which include the influence of the centrifugal force are 

If this equality is valid, then the oblate spheroid is indeed a figure of equilibrium. As follows from Equation (2.342) the frequency is defined as 

Taking into account Equation (2.327) and the equality b ja — 1/1 + e, we have 

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