Chapter IV: Front Matter (4)

45. When the liquid is bounded externally by the fixed ellipsoid
[lambda] = [lambda]1, a slight extension will give the velocity
function [phi] of the liquid in the interspace as the ellipsoid
[lambda] = 0 is passing with velocity U through the confocal position;
[phi] must now take the form x([psi] + N), and will satisfy the
conditions in the shape
_
abc /[lambda]1 abcd[lambda]
------ + | ----------------
A + B1 + C1 a1b1c1 _/ [lambda] (a² + [lambda])P
[phi] = Ux ----------------- = Ux ------------------------------------------, (1)
B0 + C0 - B1 - C1 abc /[lambda]1 abcd[lambda]
1 - ------ - | ----------------
a1b1c1 _/0 (a² + [lambda])P

and any confocal ellipsoid defined by [lambda], internal or external
to [lambda] = [lambda]1, may be supposed to swim with the liquid for
an instant, without distortion or rotation, with velocity along Ox

B_[lambda] + C_[lambda] - B1 - C1
U ----------------------------------
B0 + C0 - B1 - C1

Since - Ux is the velocity function for the liquid W´ filling the
ellipsoid [lambda] = 0, and moving bodily with it, the effective
inertia of the liquid in the interspace is

A0 + B1 + C1
----------------- W´. (2)
B0 + C0 - B1 - C1

If the ellipsoid is of revolution, with b = c,

A + 2B1
[phi] = ½Ux -------, (3)
B0 - B1

and the Stokes' current function [psi] can be written down

B - B1
[psi] = - ½Uy² -------; (4)
B0 - B1

reducing, when the liquid extends to infinity and B1 = 0, to
A B
[phi] = ½Ux --, [psi] = - ½Uy² --; (5)
B0 B0

so that in the relative motion past the body, as when fixed in the
current U parallel to xO,

/ A \ / B \
[phi]´ = ½Ux ( 1 + -- ), [psi]´ = ½Uy² ( 1 - -- ). (6)
\ B0 / \ B0 /

Changing the origin from the centre to the focus of a prolate
spheroid, then putting b² = pa, [lambda] = [lambda]´a, and proceeding
to the limit where a = [oo], we find for a paraboloid of revolution

p B p
B = ½ -------------, -- = -------------, (7)
p + [lambda]´ B0 p + [lambda]´

y²
------------- = p + [lambda]´ - 2x, (8)
p + [lambda]´

with [lambda]´ = 0 over the surface of the paraboloid; and then

[psi]´ = ½U [y² - p[root](x² + y²) + px]; (9)

[psi] = -½Up [[root](x² + y²) - x]; (10)

[phi] = -½Up log [ [root](x² + y²) + x]. (11)

The relative path of a liquid particle is along a stream line

[psi]´ = ½Uc², a constant, (12)

p²y² - (y² - c²)² p²y² - (y² - c²)²
x = -----------------, [root](x² + y²) = ----------------- (13)
2p(y² - c²) 2p(y² - c²)

a C4; while the absolute path of a particle in space will be given by

dy r - x y² - c²
-- = - ----- = -------, (14)
dx y 2py

y² - c² = a²e^(-x/p). (15)

46. Between two concentric spheres, with

a² + [lambda] = r², a² + [lambda]1 = a1², (1)

A = B = C = a³/3r³,

a³ a³ a³ a³
-- + 2 --- -- + 2 ---
r³ a1³ r³ a1³
[phi] = ½Ux -----------, [psi] = ½Uy² ----------; (2)
1 - a^4/a1² 1 - a³/a1³

and the effective inertia of the liquid in the interspace is

A0 + 2A1 a1³ + 2a³
--------- W´ = ½ --------- W´. (3)
2A0 - 2A1 a1³ - a³

When the spheres are not concentric, an expression for the effective
inertia can be found by the method of images (W. M. Hicks, _Phil.
Trans._, 1880).

The image of a source of strength [mu] at S outside a sphere of radius
a is a source of strength [mu]a/[f] at H, where OS = [f], OH = a²/f,
and a line sink reaching from the image H to the centre O of line
strength - [mu]/a; this combination will be found to produce no flow
across the surface of the sphere.

Taking Ox along OS, the Stokes' function at P for the source S is [mu]
cos PSx, and of the source H and line sink OH is [mu](a/[f]) cos PHx
and -([mu]/a)(PO - PH); so that

/ a PO - PH \
[psi] = [mu] (cos PSx + --- cos PHx - ------- ), (4)
\ [f] a /

and [psi] = -[mu], a constant, over the surface of the sphere, so that
there is no flow across.

When the source S is inside the sphere and H outside, the line sink
must extend from H to infinity in the image system; to realize
physically the condition of zero flow across the sphere, an equal sink
must be introduced at some other internal point S´.

When S and S´ lie on the same radius, taken along Ox, the Stokes'
function can be written down; and when S and S´ coalesce a doublet is
produced, with a doublet image at H.

For a doublet at S, of moment m, the Stokes' function is

d y²
m-- cos PSx = -m---; (5)
df PS³

and for its image at H the Stokes' function is

d a³ y²
m-- cos PHx = -m-- ---; (6)
df f³ PH³

so that for the combination

/a³ 1 1 \ y² / a³ f³\
[psi] = my² ( -- --- - --- ) = m-- ( --- - --- ), (7)
\f³ PH³ PS³/ f³ \PH³ PS³/

and this vanishes over the surface of the sphere.

There is ao Stokes' function when the axis of the doublet at S does
not pass through O; the image system will consist of an inclined
doublet at H, making an equal angle with OS as the doublet S, and of a
parallel negative line doublet, extending from H to O, of moment
varying as the distance from O.

A distribution of sources and doublets over a moving surface will
enable an expression to be obtained for the velocity function of a
body moving in the presence of a fixed sphere, or inside it.

The method of electrical images will enable the stream function [psi]´
to be inferred from a distribution of doublets, finite in number when
the surface is composed of two spheres intersecting at an angle
[pi]/m, where m is an integer (R. A. Herman, _Quart. Jour. of Math._
xxii.).

Thus for m = 2, the spheres are orthogonal, and it can be verified
that

/ a1³ a2³ a³ \
[psi]´ = ½Uy² ( 1 - --- - --- + -- ), (8)
\ r1³ r2³ r³ /

where a1, a2, a = a1a2/[root](a1² + a2²) is the radius of the spheres
and their circle of intersection, and r1, r2, r the distances of a
point from their centres.

The corresponding expression for two orthogonal cylinders will be

/ a1² a2² a² \
[psi]´ = Uy ( 1 - --- - --- + -- ). (9)
\ r1² r2² r² /

With a2 = [oo], these reduce to

/ 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

P = lx1 + mx2 + nx3, (6)

dP dl dm dn
-- = --x1 + --x2 + --x3
dt dt dt dt

dx1 dx2 dx3
+ l--- + m--- + n---,
dt dt dt

/ dx1 \
= l ( --- - x2R + x3Q )
\ dt /

/ dx2 \
+ m ( --- - x3P + x1R )
\ dt /

/ dx3 \
+ n ( --- - x1Q + x2P )
\ dt /

= lX + mY + nZ, (7)

for all values of l, m, n.

Next, taking a fixed origin [Omega] and axes parallel to Ox, Oy, Oz
through O, and denoting by x, y, z the coordinates of O, and by G the
component angular momentum about [Omega] in the direction (l, m, n)

G = l(y1 - x2z + x3y)
+ m(y2 - x3x + x1z)
+ n(y3 - x1y + x2x). (8)

Differentiating with respect to t, and afterwards moving the fixed
origin up to the moving origin O, so that

dx dy dz
x = y = z = 0, but -- = U, -- = V, -- = W,
dt dt dt

dG / dy1 \
-- = l ( --- - y2R + y3Q - x2W + x3V )
dt \ dt /

/ dy2 \
+ m ( --- - y3P + y1R - x3U + x1W )
\ dt /

/ dy3 \
+ n ( --- - y1Q + y2P - x1V + x2U )
\ dt /

= lL + mM + nN, (9)

for all values of l, m, n.

When no external force acts, the case which we shall consider, there
are three integrals of the equations of motion

(i.) T = constant,

(ii.) x1² + x2² + x3² = F², a constant,

(iii.) x1y1 + x2y2 + x3y3 = n = GF, a constant;

and the dynamical equations in (3) express the fact that x1, x2, x3
are the components of a constant vector having a fixed direction;
while (4) shows that the vector resultant of y1, y2, y3 moves as if
subject to a couple of components

x2W - x3V, x3U - x1W, x1V - x2U, (10)

and the resultant couple is therefore perpendicular to F, the
resultant of x1, x2, x3, so that the component along OF is constant,
as expressed by (iii).

If a fourth integral is obtainable, the solution is reducible to a
quadrature, but this is not possible except in a limited series of
cases, investigated by H. Weber, F. Kötter, R. Liouville, Caspary,
Jukovsky, Liapounoff, Kolosoff and others, chiefly Russian
mathematicians; and the general solution requires the double-theta
hyperelliptic function.

49. In the motion which can be solved by the elliptic function, the
most general expression of the kinetic energy was shown by A. Clebsch
to take the form

T = ½p(x1² + x2²) + ½p´x3²
+ q(x1y1 + x2y2) + q´x3y3
+ ½r(y1² + y2²) + ½r´y3² (1)

so that a fourth integral is given by

dy3/dt = 0, y3 = constant; (2)

dx3
--- = x1(qx2 + ry2) - x2(qx1 + ry1) = r(x1y2 - x2y1), (3)
dt

1 / dx3 \²
-- ( --- ) = (x1² + x2²)(y1² + y2²) - (x1y1 + x2y2)²
r² \ dt /

= (x1² + x2²)(y1² + y2²) - (FG - x3y3)²
= (x1² + x2²)(y1² + y2² + y3² - G²) - (Gx3 - Fy3)², (4)

in which

x1² + x2² = F² - x3², x1y1 + x2y2 = FG - x3y3, (5)

r(y1² + y2²) = 2T - p(x1² + x2²) - p´x3²
- 2q(x1y1 + x2y2) - 2q´x3y3 - r´y3²
= (p - p´)x3² + 2(q - q´)x3y3 + m1, (6)

m1 - 2T - pF² - 2qFG - r1y3² (7)

so that

1 / dx3 \²
-- ( --- ) = X3 (8)
r² \ dt /

where X3 is a quartic function of x3, and thus t is given by an
elliptic integral of the first kind; and by inversion x3 is in
elliptic function of the time t. Now

(x1 - x2i)(y1 + y2i) = x1y1 + x2y2 + i(x1y2 - x2y1)
= FG - xy3y3 + i[V-]X3, (9)

y1 + y2i FG - x3y3 + i[root]X3
-------- = --------------------- , (10)
x1 + x2i x1² + x2²

d
-- (x1 + x2i) = -i[(q´ - q)x3 + r´y3] + irx3(y1 + y2i), (11)
dt

d FG - x3y3 + i[root]X3
--- log (x1 + x2i) = dti -(q´ - q)x - r´y + rx ---------------------, (12)
dti F² - x3²

d /x1 + x2i Fy3 - Gx3
--- log \ / -------- = -(q´ - q)x3 - (r´ - r)y3 - Fr---------, (13)
dti \/ x1 - x2i F² - x3²

requiring the elliptic integral of the third kind; thence the
expression of x1 + x2i and y1 + y2i.

Introducing Euler's angles [theta], [phi], [psi],

x1 = F sin [theta] sin [phi], x2 = F sin [theta] cos [phi],

x1 + x2i = iF sin [theta][epsilon]^(-[psi]i), x3 = F cos [theta]; (14)

d[psi]
sin [theta] ------ = P sin [phi] + Q cos[phi], (15)
dt

d[psi] dT dT
F sin²[theta] ------ = --- x1 + --- x2
dt dy1 dy2

= (qx1 + ry1)x1 + (qx2 + ry2)x2

= qx1² + x2²) + r (x1y1 + x2y2)

= gF² sin² [theta] + r(FG - x3y3), (16)
_
/ FG - x3y3 Fr dx3
[psi] - qFt = | --------- --------, (17)
_/ F² - x3² [root]X3

elliptic integrals of the third kind.

Employing G. Kirchhoff's expressions for X, Y, Z, the coordinates of
the centre of the body,
__ __ __
FX = y1 cos xY + y2 cos yY + y3 cos zY, (18)
__ __ __
FY = -y1 cos xX + y2 cos yX + y3 cos zX, (19)
__ __ __
G = y1 cos xZ + y2 cos yZ + y3 cos zZ, (20)

F²(X² + Y²) = y1² + y2² + y3² - G², (21)

Fy3 - Gx3 + i[root]X3
F(X + Yi) = --------------------- [epsilon]^[psi]_i. (22)
[root](F² - x3²)

Suppose x3 - F is a repeated factor of X3, then y3 = G, and
_ _
| p´ - p q´ - q |
X3 = (x3 - F)² | ------(x3 + F)² + 2------G(x3 + F) - G² |, (23)
|_ r r _|

and putting x3 - F = y,
_
/ dy \² | p´ - p q´ - q
( -- ) = r²y² | 4 ------ F² + 4 ------ FG - G²
\ dt / |_ r r
_
/ p´ - p q´ - q \ p´ - p |
+ 2 ( 2 ------ F + ------ G ) y + ------ y² |, (24)
\ r r / r _|

so that the stability of this axial movement is secured if

p´ - p q´ - q
A = 4 ------F² + 4 ------FG - G² (25)
r r

is negative, and then the axis makes r[V-](-A)/[pi] nutations per
second. Otherwise, if A is positive
_
/ dy
rt = | ----------------------
_/ y[root](A + 2By + Cy²)

1 sh^(-1) [root]A[root](A + 2By + Cy²) 1 ch^(-1) A + By
= ------- ---------------------------- = ------- ---------------, (26)
[root]A ch^(-1) y[root](B² ~ AC) [root]A sh^(-1) y[root](B² ~ AC)

and the axis falls away ultimately from its original direction.

A number of cases are worked out in the American Journal of
Mathematics (1907), in which the motion is made algebraical by the use
of the pseudo-elliptic integral. To give a simple instance, changing
to the stereographic projection by putting tan ½[theta] = x,

(Nxe[psi]i)^3/2 = (x + 1)[root]X1 + i(x - 1)[root]X2, (27)

X1
-- = ±ax^4 + 2ax³ ± 3(a + b)x² + 2bx ± b, (28)
X2

N³ = -8(a + b), (29)

will give a possible state of motion of the axis of the body; and the
motion of the centre may then be inferred from (22).

50. The theory preceding is of practical application in the investigation of the stability of the axial motion of a submarine boat, of the elongated gas bag of an airship, or of a spinning rifled projectile. In the steady motion under no force of such a body in a medium, the centre of gravity describes a helix, while the axis describes a cone round the direction of motion of the centre of gravity, and the couple causing precession is due to the displacement of the medium.

In the absence of a medium the inertia of the body to translation is the same in all directions, and is measured by the weight W, and under no force the C.G. proceeds in a straight line, and the axis of rotation through the C.G. preserves its original direction, if a principal axis of the body; otherwise the axis describes a cone, right circular if the body has uniaxial symmetry, and a Poinsot cone in the general case.

But the presence of the medium makes the effective inertia depend on the direction of motion with respect to the external shape of the body, and on W´ the weight of fluid medium displaced.

Consider, for example, a submarine boat under water; the inertia is
different for axial and broadside motion, and may be represented by

c1 = W + W´[alpha], c2 = W + W´[beta], (1)

where [alpha], [beta] are numerical factors depending on the external
shape; and if the C.G. is moving with velocity V at an angle [phi]
with the axis, so that the axial and broadside component of velocity
is u = V cos [phi], v = V sin [phi], the total momentum F of the
medium, represented by the vector OF at an angle [theta] with the
axis, will have components, expressed in sec. lb.,

u V v V
F cos [theta] = c1 --- = (W + W´[alpha]) --- cos [phi], F sin [theta] = c2 --- = (W + W´[beta]) --- sin [phi]. (2)
g g g g

Suppose the body is kept from turning as it advances; after t seconds
the C.G. will have moved from O to O´, where OO´ = Vt; and at O´ the
momentum is the same in magnitude as before, but its vector is
displaced from OF to O´F´.

For the body alone the resultant of the components of momentum