[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. Inside a cylinder [phi]1 + [psi]1i = -½iR(x + yi)²a²/(a² + b²), (10) [phi]2 + [psi]2i = ½iR(x + yi)²b²/(a² + b²), (11) and for the interspace, the ellipse [alpha] being fixed, and [alpha]1 revolving with angular velocity R [phi]1 + [psi]1i = -1/8 iRc²sh 2([eta] - [alpha] + [xi]i)(ch 2[alpha] + 1)/sh 2([alpha]1 - [alpha]), (12) [phi]2 + [psi]2i = 1/8 iRc²sh 2([eta] - [alpha] + [xi]i)(ch 2[alpha] - 1)/sh 2([alpha]1 - [alpha]), (13) satisfying the condition that [psi]1 and [psi]2 are zero over [eta] = [alpha], and over [eta] = [alpha]1 [psi]1 + ½Rx² = 1/8 Rc²(ch 2[alpha]1 + 1), (14) [psi]2 + ½Ry² = 1/8 Rc²(ch 2[alpha]1 - 1), (15) constant values. In a similar way the more general state of motion may be analysed, given by w = m ch 2([zeta] - [gamma]), [gamma] = [alpha] + [beta]i, (16)