(4[pi][rho]A + [alpha])x² = (4[pi][rho]B + [beta])b²
= (4[pi][rho]C + [gamma])c², (16)
the surfaces of equal pressure are similar to the external case, which
can then be removed without affecting the motion, provided [alpha],
[beta], [gamma] remain constant.
This is so when the axis of revolution is a principal axis, say Oz;
when
[Omega]1 = 0, [Omega]2 = 0, [xi] = 0, [eta] = 0. (17)
If [Omega]3 = 0 or [theta]3 = [zeta] in addition, we obtain the
solution of Jacobi's ellipsoid of liquid of three unequal axes,
rotating bodily about the least axis; and putting a = b, Maclaurin's
solution is obtained of the rotating spheroid.
In the general motion again of the liquid filling a case, when a = b,
[Omega]3 may be replaced by zero, and the equations, hydrodynamical
and dynamical, reduce to
d[xi] 2c² d[eta] 2a²
----- = - -------[Omega]2 [zeta], ------ = -------[Omega]1 [zeta],
dt a² + c² dt a² + c²
d[zeta] 2c²
------- = -------([Omega]2 [xi] - [Omega]2 [eta]) (18)
dt a² + c²
d[Omega]1 a² + c²
--------- = [Omega]2 [zeta] + -------[eta][zeta],
dt a² - c²
d[Omega]2 a² + c²
--------- = [Omega]1 [zeta] + -------[xi][zeta]; (19)
dt a² - c²
of which three integrals are
a²
[xi]² + [eta]² = L - --[zeta]², (20)
c²
(a² + c²)²
[Omega]1² + [Omega]2² = M + ------------ [zeta]², (21)
2c²(a² - c²)
a² + c²
[Omega]1 [xi] + [Omega]2 [eta]N = + ------- [zeta]²; (22)
4c²
and then
/ d[zeta]\² 4c^4
( ------- ) = --------- ([Omega]2[xi] - [Omega]1²[eta])²
\ dt / (a² + c²)
4c^4
= ---------- [([xi]² + [eta]²)([Omega)1² + [Omega]2²) -([Omega]1[xi] + [Omega]2[eta])²]
(a² + c²)²
_
4c^4 | / (a² + c²)² a² (a² + c²)\
= ---------- | LM - N² + ( L------------ - M --- - N --------- ) [zeta]²
(a² + c²)² |_ \ 2c²(a² + c²) c² 2c² /
_
(a² + c²)(9a² - c²) |
- ------------------- [zeta]^4 | = Z, (23)
16c^4(a² - c²) _|
where Z is a quadratic in [zeta]², so that [zeta] is an elliptic
function of t, except when c = a, or 3a.
Put [Omega]1 = [Omega] cos [phi], [Omega]2 = -[Omega] sin [phi],
d[phi] d[Omega]1 d[Omega]2 (a² + c²)
[Omega]2 ------ = -----------[Omega]2 - [Omega]1 --------- = [Omega]²[zeta] - ---------([Omega]1 [xi] + [Omega]2 [eta])[zeta], (24)
dt dt dt (a² - c²)
a² + c²
N + -------
d[phi] (a² + c²) 4c²
------ = [zeta] - --------- · -------------------------, (25)
dt (a² - c²) (a² + c²)²
M + ------------[zeta]²
2c²(a² - c²)
a² + c²
_ _ N + -------[zeta]²
/ [zeta]d a² + c² / 4c² [zeta] d[zeta]
[phi] = | ------- - ------- | ------------------------ · --------------, (26)
_/ [root]Z a² - c² _/ (a² + c²)² [root]Z
M + ------------[zeta]²
2c²(a² - c²)
which, as Z is a quadratic function of [zeta]², are non-elliptic
integrals; so also for [psi], where [xi] = [omega] cos [psi], [eta] =
-[omega] sin [psi].