/ a^5 \ x / a^4 \ x
[psi]´ = ½Uy² ( 1 - --- ) ---, or Uy ( 1 - --- ) ---, (10)
\ r^5 / a \ r^4 / a
for a sphere or cylinder, and a diametral plane.
Two equal spheres, intersecting at 120°, will require
_ _
| x a³ a^4(a - 2x) a³ a^4(a + 2x) |
[psi]´ = ½Uy² | --- - ---- + ----------- + ---- - ----------- |, (11)
|_ a 2r1³ 2r1^5 2r2³ 2r2^5 _|
with a similar expression for cylinders; so that the plane x = 0 may
be introduced as a boundary, cutting the surface at 60°. The motion of
these cylinders across the line of centres is the equivalent of a line
doublet along each axis.
47. The extension of Green's solution to a rotation of the ellipsoid
was made by A. Clebsch, by taking a velocity function
[phi] = xy[chi] (1)
for a rotation R about Oz; and a similar procedure shows that an
ellipsoidal surface [lambda] may be in rotation about Oz without
disturbing the motion if
/ 1 1 \ dx
( ------------ + ------------ ) [chi] + 2---------
\ a² + [lambda] b² + [lambda] / d[lambda]
R = - -----------------------------------------------------, (2)
1/(b² + [lambda] - 1/(a² = [lambda])
and that the continuity of the liquid is secured if
d[chi]
(a² + [lambda])^3/2 (b² + [lambda])^3/2 (c² + [lambda]) ½--------- = constant, (3)
d[lambda]
_
/ [oo] Nd[lambda] N B_[lambda] - A_[lambda]
[chi] = | ------------------------------- = --- . -----------------------; (4)
_/[lambda] (a² + [lambda])(b² + [lambda])P abc a² - b²
and at the surface [lambda] = 0,
/ 1 1\ N B0 - A0 N 1
( -- + -- ) --- ------- - --- ----
\a² b²/ abc a² - b² abc a²b²
R = - ----------------------------------, (5)
1/b² - 1/a²
N 1/b² - 1/a²
--- = R --------------------------, (6)
abc 1 / 1 1\ B0 - A0
---- - ( -- + -- ) -------
a²b² \a² b²/ a² - b²
(a² - b²)²/(a² + b²)
= R -------------------------------.
(a² - b²)/(a² + b²) - (B0 - A0)
The velocity function of the liquid inside the ellipsoid [lambda] = 0
due to the same angular velocity will be
[phi]1 = Rxy(a² - b²)/(a² + b²), (7)
and on the surface outside
N B0 - A0
[phi]0 = xy[chi]0 = xy--- -------, (8)
abc a² - b²
so that the ratio of the exterior and interior value of [phi] at the
surface is
[phi]0 B0 - A0
------ = -------------------------------, (9)
[phi]1 (a² - b²)/(a² + b²) - (B0 - A0)
and this is the ratio of the effective angular inertia of the liquid,
outside and inside the ellipsoid [lambda] = 0.
The extension to the case where the liquid is bounded externally by a
fixed ellipsoid [lambda] = [lambda]1 is made in a similar manner, by
putting
[phi] = xy([chi] + M), (10)
and the ratio of the effective angular inertia in (9) is changed to
a1² - b1² abc
(B0 - A0) - (B1 - A1) + --------- ------
a1² + b1² a1b1c1
--------------------------------------------------. (11)
a² - b² a1² - b1² abc
------- - --------- ------ - (B0 - A0) + (B1 - A1)
a² + b² a1² + b1² a1b1c1
Make c = [oo] for confocal elliptic cylinders; and then
_
/[oo] ab ab / /b² + [lambda] \
A[lambda] = | ----------------------------------------------------- = ------- ( 1 - / ------------- ), (12)
_/[lambda] (a² + [lambda])[root]([4·a² + [lambda]·b² + [lambda]) a² - b² \ \/ a² + [lambda] /
ab / /a² + [lambda] \
B[lambda] = ------- ( / ------------- - 1 ), C[lambda]= 0;
a² - b² \ \/ b² + [lambda] /
and then as above in § 31, with
a = c ch [alpha], b = c sh [alpha],
a1 = [root](a² + [lambda]) = c ch [alpha]1, b1 = c sh [alpha]1 (13)
the ratio in (11) agrees with § 31 (6).
As before in § 31, the rotation may be resolved into a shear-pair, in
planes perpendicular to Ox and Oy.
A torsion of the ellipsoidal surface will give rise to a velocity
function of the form [phi] = xyz[Omega], where [Omega] can be
expressed by the elliptic integrals A_[lambda], B_[lambda],
C_[lambda], in a similar manner, since
_
/ [oo]
[Omega] = L | d[lambda]/P³
_/ [lambda]
48. The determination of the [phi]'s and [chi]'s is a kinematical
problem, solved as yet only for a few cases, such as those discussed
above.
But supposing them determined for the motion of a body through a
liquid, the kinetic energy T of the system, liquid and body, is
expressible as a quadratic function of the components U, V, W, P, Q,
R. The partial differential coefficient of T with respect to a
component of velocity, linear or angular, will be the component of
momentum, linear or angular, which corresponds.
Conversely, if the kinetic energy T is expressed as a quadratic
function of x1, x2, x3, y1, y2, y3, the components of momentum, the
partial differential coefficient with respect to a momentum component
will give the component of velocity to correspond.
These theorems, which hold for the motion of a single rigid body, are
true generally for a flexible system, such as considered here for a
liquid, with one or more rigid bodies swimming in it; and they express
the statement that the work done by an impulse is the product of the
impulse and the arithmetic mean of the initial and final velocity; so
that the kinetic energy is the work done by the impulse in starting
the motion from rest.
Thus if T is expressed as a quadratic function of U, V, W, P, Q, R,
the components of momentum corresponding are
dT dT dT
x1 = --, x2 = --, x3 = --, (1)
dU dV dW
dT dT dT
y1 = --, y2 = --, y3 = --;
dP dQ dR
but when it is expressed as a quadratic function of x1, x2, x3, y1,
y2, y3,
dT dT dT
U = ---, V = ---, W = ---, (2)
dx1 dx2 dx3
dT dT dT
P = ---, Q = ---, R = ---.
dy1 dy2 dy3
The second system of expression was chosen by Clebsch and adopted by
Halphen in his _Fonctions elliptiques_; and thence the dynamical
equations follow
dx1 dT dT
X = --- - x2--- + x3---, Y = ..., Z = ..., (3)
dt dy3 dy2
dy1 dT dT dT dT
L = --- - y2--- + y3--- - x2--- + x2---, M = ..., N = ..., (4)
dt dy3 dy2 dx3 dx2
where X, Y, Z, L, M, N denote components of external applied force on
the body.
These equations are proved by taking a line fixed in space, whose
direction cosines are l, m, n, then
dl dm dn
-- = mR - nQ, -- = nP - lR, -- = lQ - mP. (5)
dt dt dt
If P denotes the resultant linear impulse or momentum in this
direction