(5)
with
∇2ψ1′ = −2R = −2m1 (1/a2 + 1/b2),
ψ1 = m1 (x2/a2 + y2/b2) − ½R (x2 + y2)
= −½R (x2 − y2) (a2 − b2) / (a2 + b2),
(6)
φ1 = Rxy (a2 − b2) / (a2 + b2),
w1 = φ1 + ψ1i = −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)