On the shape of a rotating liquid surface

A differential equation for the top surface of a liquid rotating in a cylinder is derived. Pressure difference at the sides of this curved surface due to its tension is taken into account. The derived equation is solved analytically in an approximation of small angular velocities. The results show that the shape of the surface is not exactly paraboloidal, as is usually considered. More precisely, it is almost paraboloidal, but only if the cylinder's radius is relatively large. Deviation from the paraboloid is then concentrated mainly around the rim of the cylinder. When the cylinder's radius is relatively small the surface of rotating liquid is flatter than paraboloid.