ψ′ = ψ + ½R (x2 + y2) = [ −m sh 2 (η − α) + ¼Rc2 ] cos 2ξ + ¼Rc2 ch 2η,
(2)
which is constant over the ellipse η if
¼ Rc2 = m sh 2 (η − α);
(3)
so that this ellipse can be rotating with this angular velocity R for an instant without distortion, the ellipse α being fixed.
For the liquid filling the interior of a rotating elliptic cylinder of cross section
x2/a2 + y2/b2 = 1,
(4)
ψ1′ = m1 (x2/a2 + y2/b2)