Equation of equilibrium and level surfaces

In a frame of reference rotating together with the fluid, a particle is at rest and thus the sum of three forces is equal to zero  

Here p is the pressure, ga the field of attraction, 5 the density of the fluid, and r the vector directed away from the axis of rotation and it is equal in magnitude to the distance between a particle and this axis. The first two terms of Equation (2.332) characterize the real forces acting on the particle, namely the surface and attraction ones. At the same time the last term is a centrifugal force, and it is introduced because we consider a non-inertial frame of reference. It is convenient to represent Equation (2.332) as 

This elementary transformation of Equation (2.332) allows us to visualize better geometry of the force f and level surface of its potential. Assume that the surface of 

If the pressure is constant on the fluid surface 5, then, as follows from Equation (2.334), for any two points on this surface located very close to each other we have 

in equilibrium the level surfaces of the potential serve as surfaces of equal pressure this is not surprising because the potential of the gravitational field and pressure may differ by a constant only. Inasmuch as the force f and grad 5 are parallel to each other then we have 

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