[phi] = m ch 2([eta] - [alpha]) sin 2[xi],
[psi] = -m sh 2([eta] - [alpha]) cos 2[xi]; (1)
in which [psi] = 0 over the ellipse [alpha], and
[psi]´ = [psi] + ½R(x² + y²)
= [-m sh 2([eta] - [alpha]) + ¼Rc²] cos 2[xi] + ¼Rc² ch 2[eta], (2)
which is constant over the ellipse [eta] if
¼Rc² = m sh 2([eta] - [alpha]); (3)
so that this ellipse can be rotating with this angular velocity R for
an instant without distortion, the ellipse [alpha] being fixed.
For the liquid filling the interior of a rotating elliptic cylinder of
cross section
x²/a² + y²/b² = 1, (4)
[psi]1´ = m1(x²/a² + y²/b²) (5)
with
[nabla]²[psi]1´ = -2R = -2m1(1/a² + 1/b²),
[psi]1 = m1(x²/a² + y²/b²) - ½R(x² + y²)
= -½R(x² - y²)(a² - b²)/(a² + b²), (6)
[phi]1 = Rxy(a² - b²)/(a² + b²),
w1 = [phi]1 + [psi]1i = -½iR(x + yi)²(a² - b²)/(a² + b²).
The velocity of a liquid particle is thus (a² - b²)/(a² + b²) of what
it would be if the liquid was frozen and rotating bodily with the
ellipse; and so the effective angular inertia of the liquid is (a² -
b²)²/(a² + b²)² of the solid; and the effective radius of gyration,
solid and liquid, is given by
k² = ¼(a² + b²), and ¼(a² - b²)²/(a² + b²). (7)
For the liquid in the interspace between [alpha] and [eta],
[phi] m ch 2([eta] - [alpha]) sin 2[xi]
------ = -------------------------------------------
[phi]1 ¼Rc² sh 2[eta] sin 2[xi](a² - b²)/(a² + b²)
= 1/th 2([eta] - [alpha])th 2[eta]; (8)
and the effective k² of the liquid is reduced to
¼c²/th 2([eta] - [alpha]) sh 2[eta], (9)
which becomes ¼c²/sh 2[eta] = 1/8 (a² - b²)/ab, when [alpha] = [oo],
and the liquid surrounds the ellipse [eta] to infinity.
An angular velocity R, which gives components -Ry, Rx of velocity to a
body, can be resolved into two shearing velocities, -R parallel to Ox,
and R parallel to Oy; and then [psi] is resolved into [psi]1 + [psi]2,
such that [psi]1 + ½Rx² and [psi]2 + ½Ry² is constant over the
boundary.