Encyclopaedia Britannica, 11th Edition, "Hydromechanics" to "Ichnography"

Chapter III: Front Matter (3)

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by Green's transformation, d[nu] denoting an elementary step along the
normal to the exterior of the surface; so that d[phi]/d[nu] = 0 over
the surface makes T = 0, and then

/d[phi]\² /d[phi]\² /d[phi]\² d[phi] d[phi] d[phi]
( ------ ) + ( ------ ) + ( ------ ) = 0, ------ = 0, ------ = 0, ------ = 0 (2)
\ dx / \ dy / \ dz / dx dy dz

If the actual motion at any instant is supposed to be generated
instantaneously from rest by the application of pressure impulse over
the surface, or suddenly reduced to rest again, then, since no natural
forces can act impulsively throughout the liquid, the pressure impulse
[~[omega]] satisfies the equations

1 d[~omega] 1 d[~omega] 1 d[~omega]
----- --------- = -u, ----- --------- = -v, ----- --------- = -[~omega], (3)
[rho] dx [rho] dy [rho] dz

[~omega] = [rho][phi] + a constant, (4)

and the constant may be ignored; and Green's transformation of the
energy T amounts to the theorem that the work done by an impulse is
the product of the impulse and average velocity, or half the velocity
from rest.

In a multiply connected space, like a ring, with a multiply valued
velocity function [phi], the liquid can circulate in the circuits
independently of any motion of the surface; thus, for example,

[phi] = m[theta] = m tan^(-1) y/x (5)

will give motion to the liquid, circulating in any ring-shaped figure
of revolution round Oz.

To find the kinetic energy of such motion in a multiply connected
space, the channels must be supposed barred, and the space made
acyclic by a membrane, moving with the velocity of the liquid; and
then if k denotes the cyclic constant of [phi] in any circuit, or the
value by which [phi] has increased in completing the circuit, the
values of [phi] on the two sides of the membrane are taken as
differing by k, so that the integral over the membrane
_ _ _ _
/ / d[phi] / / d[phi]
| | [phi] ------ dS = k | | ------ dS, (6)
_/_/ d[nu] _/_/ d[nu]

and this term is to be added to the terms in (1) to obtain the
additional part in the kinetic energy; the continuity shows that the
integral is independent of the shape of the barrier membrane, and its
position. Thus, in (5), the cyclic constant k = 2[pi]m.

For example, in the equilateral triangle of (8) § 28, referred to
coordinate axes made by the base and height,

[psi]´ = -2R[alpha][beta][gamma]/h = -½Ry[(h - y)² - 3x²]/h (8)

[psi] = [psi]´ - ½R [(1/3h - y)² + x²]

= -½R [½h³ + 1/3 h²y + h) (x² - y²) - 3x²y + y³] /h (9)

and over the base y = 0,

dx/d[nu] = -dx/dy = + ½R(1/3 h² - 3x²)/h, [psi] = -½R(1/9 h² + x²). (10)

Integrating over the base, to obtain one-third of the kinetic energy
T,
_
/ h/[root]3
1/3 T = ½[rho] | ¼R²(3x^4 - 1/27 h^4) dx/h
_/ -h/[root]3

= [rho]R²h^4/135[root]3 (11)

so that the effective k² of the liquid filling the triangle is given
by

k² = T/½[rho]R²A = 2h²/45

= 2/5 (radius of the inscribed circle)², (12)

or two-fifths of the k² for the solid triangle.

Again, since

d[phi]/d[nu] = d[psi]/ds, d[phi]/ds = -d[psi]/d[nu], (13)
_ _
/ /
T = ½[rho] | [phi] d[psi] = -½[rho] | [psi] d[phi]. (14)
_/ _/

With the Stokes' function [psi] for motion symmetrical about an
axis.
_ _
/ d[psi] /
T = ½[rho] | [phi] ------ 2[pi]y ds = [pi][rho] | [phi] d[psi]. (15)
_/ yds _/

37. _Flow, Circulation, and Vortex Motion._--The line integral of the
tangential velocity along a curve from one point to another, defined
by
_ _
/ / dx dy dz \ /
| ( u-- + v-- + w-- ) ds = | (u dx + v dy + z dz), (1)
_/ \ ds ds ds / _/

is called the "flux" along the curve from the first to the second
point; and if the curve closes in on itself the line integral round
the curve is called the "circulation" in the curve.

With a velocity function [phi], the flow
_
/
- | d[phi] = [phi]1 - [phi]2, (2)
_/

so that the flow is independent of the curve for all curves mutually
reconcilable; and the circulation round a closed curve is zero, if the
curve can be reduced to a point without leaving a region for which
[phi] is single valued.

If through every point of a small closed curve the vortex lines are
drawn, a tube is obtained, and the fluid contained is called a _vortex
filament_.

By analogy with the spin of a rigid body, the component spin of the
fluid in any plane at a point is defined as the circulation round a
small area in the plane enclosing the point, divided by twice the
area. For in a rigid body, rotating about Oz with angular velocity
[zeta], the circulation round a curve in the plane xy is
_
/ / dy dx \
| [zeta] ( x -- - y -- ) ds = [zeta] times twice the area. (3)
_/ \ ds ds /

In a fluid, the circulation round an elementary area dxdy is equal to

/ dv \ / du \ / dv du \
udx + ( v + --dx )dy - ( u + --dy )dx - vdy = ( -- - -- )dx dy, (4)
\ dx / \ dy / \ dx dy /

so that the component spin is

/ dv du \
½ ( -- - -- ) = [zeta], (5)
\ dx dy /

in the previous notation of § 24; so also for the other two components
[xi] and [eta].

Since the circulation round any triangular area of given aspect is the
sum of the circulation round the projections of the area on the
coordinate planes, the composition of the components of spin, [xi],
[eta], [zeta], is according to the vector law. Hence in any
infinitesimal part of the fluid the circulation is zero round every
small plane curve passing through the vortex line; and consequently
the circulation round any curve drawn on the surface of a vortex
filament is zero.

If at any two points of a vortex line the cross-section ABC, A´B´C´ is
drawn of the vortex filament, joined by the vortex line AA´, then,
since the flow in AA´ is taken in opposite directions in the complete
circuit ABC AA´B´C´ A´A, the resultant flow in AA´ cancels, and the
circulation in ABC, A´B´C´ is the same; this is expressed by saying
that at all points of a vortex filament [omega][alpha] is constant
where [alpha] is the cross-section of the filament and [omega] the
resultant spin (W. K. Clifford, _Kinematic_, book iii.).

So far these theorems on vortex motion are kinematical; but
introducing the equations of motion of § 22,

Du dQ Dv dQ Dw dQ
-- + -- = 0, -- + -- = 0, -- + -- = 0, (6)
dt dx dt dy dt dz
_
/
Q = | dp/[rho] + V, (7)
_/

and taking dx, dy, dz in the direction of u, v, w, and

dx : dy : dz = u : v : w,

D / \ Du D dx
-- (u dx + v dy + w dz ) = -- dx + u ---- + ... = -dQ + ½dq², (8)
dt \ / dt dt

and integrating round a closed curve
_
D /
-- | (u dx + v dy + w dz) = 0, (9)
dt _/

and the circulation in any circuit composed of the same fluid
particles is constant; and if the motion is differential irrotational
and due to a velocity function, the circulation is zero round all
reconcilable paths. Interpreted dynamically the normal pressure of the
surrounding fluid on a tube cannot create any circulation in the tube.

The circulation being always zero round a small plane curve passing
through the axis of spin in vortical motion, it follows conversely
that a vortex filament is composed always of the same fluid particles;
and since the circulation round a cross-section of a vortex filament
is constant, not changing with the time, it follows from the previous
kinematical theorem that [alpha][omega] is constant for all time, and
the same for every cross-section of the vortex filament.

A vortex filament must close on itself, or end on a bounding surface,
as seen when the tip of a spoon is drawn through the surface of water.

Denoting the cross-section [alpha] of a filament by dS and its mass by
dm, the quantity [omega]dS/dm is called the _vorticity_; this is the
same at all points of a filament, and it does not change during the
motion; and the vorticity is given by [omega] cos[epsilon]dS/dm, if dS
is the oblique section of which the normal makes an angle [epsilon]
with the filament, while the aggregate vorticity of a mass M inside a
surface S is
_
/
M^(-1) | [omega] cos [epsilon] dS.
_/

Employing the equation of continuity when the liquid is homogeneous,

/ d[zeta] d[eta]\ d² d² d²
2( ------ - ------ ) = [nabla]²u, ... , [nabla]² = - --- - --- - ---, (10)
\ dy dz / dx² dy² dz²

which is expressed by

[nabla]²(u,v,w) = 2 curl ([xi], [eta], [zeta]),
([xi], [eta], [zeta]) = ½ curl (u, v, w). (11)

38. _Moving Axes in Hydrodynamics._--In many problems, such as the
motion of a solid in liquid, it is convenient to take coordinate axes
fixed to the solid and moving with it as the movable trihedron frame
of reference. The components of velocity of the moving origin are
denoted by U, V, W, and the components of angular velocity of the
frame of reference by P, Q, R; and then if u, v, w denote the
components of fluid velocity in space, and u´, v´, w´ the components
relative to the axes at a point (x, y, z) fixed to the frame of
reference, we have

u = U + u´ - yR + zQ, (1)
v = V + v´- zP + xR,
w = W + w´ - xQ + yP.

Now if k denotes the component of absolute velocity in a direction
fixed in space whose direction cosines are l, m, n,

k = lu + mv + nw; (2)

and in the infinitesimal element of time dt, the coordinates of the
fluid particle at (x, y, z) will have changed by (u´, v´, w´)dt; so
that

Dk dl dm dn
-- = --u + --v + --w
dt dt dt dt

/ du du du du \
+ l( -- + u´-- + v´-- + w´-- )
\ dt dx dy dz /

/ dv dv dv dv \
+ m( -- + u´-- + v´-- + w´-- )
\ dt dx dy dz /

/ dw dw dw dw \
+ n( -- + u´-- + v´-- + w´-- ). (3)
\ dt dx dy dz /

But as l, m, n are the direction cosines of a line fixed in space,

dl dm dn
-- = mR - nQ, -- = nP - lR, -- = lQ - mP; (4)
dt dt dt

so that

Dk / du du du du \
-- = l( -- - vR + wQ + u´-- + v´-- + w´-- ) + m(...) + n(...)
dt \ dt dx dy dz /

/ 1 dp \ / 1 dp \ / 1 dp \
= l( X- --- -- ) + m( Y - --- -- ) + n( Z - --- -- ), (5)
\ p dx / \ p dy / \ p dz /

for all values of l, m, n, leading to the equations of motion with
moving axes.

When the motion is such that

d[phi] d[psi] d[phi] d[psi] d[phi] d[psi]
u = - ------ - m------, v = - ------ - m------, w = - ------ - m------, (6)
dx dx dy dy dz dz

as in §25 (1), a first integral of the equations in (5) may be written
_
/ dp d[phi] d[psi] / d[phi] d[psi] \
| ----- + V + ½q² - ------ - m------ + (u - u´) ( ------ + m------ )
_/ [rho] dt dt \ dx dx /

/ d[phi] d[psi] \ / d[phi] d[psi] \
+ (v - v´)( ------ + m------ ) + (w - w´)( ------ + m------ ) = F(t), (7)
\ dy dy / \ dz dz /
in which

d[phi] d[phi] d[phi] d[phi]
------ - (u - u´)------ - (v -v´)------ - (w - w´)------
dt dx dy dz

d[phi] d[phi] d[phi] d[phi]
= ------ - (U - yR + zQ)------ - (V - zP + xR)------ - (W - xQ + yP)------ (8)
dt dx dy dz

is the time-rate of change of [phi] at a point fixed in space, which
is left behind with velocity components u - u´, v - v´, w - w´.

In the case of a steady motion of homogeneous liquid symmetrical about
Ox, where O is advancing with velocity U, the equation (5) of § 34

p/[rho] + V + ½q´² - [f]([psi]´) = constant (9)

becomes transformed into

p U d[psi]
----- + V + ½q² - --- ------ + ½U² - [f]([psi] + ½Uy²) = constant, (10)
[rho] y dy

[psi]´ = [psi] + ¼U², (11)

subject to the condition, from (4) §34,

y^(-2)[nabla]²[psi]´ = -[f]´([psi]´),
y^(-2)[nabla]²[psi] = -[f]´([psi] + ½Uy²). (12)

Thus, for example, with

[psi]´ = ¾Uy²(r²a^(-2) - 1), r² = x² + y², (13)

for the space inside the sphere r = a, compared with the value of
[psi]´ in §34 (13) for the space outside, there is no discontinuity of
the velocity in crossing the surface.

Inside the sphere

d / 1 d[psi]´\ d / 1 d[psi]´\ 15 y
2[zeta] = --- ( --- ------- ) + --- ( --- ------- ) = ---U ---, (14)
dx \ y dx / dy \ y dy / 2 a²

so that §34 (4) is satisfied, with

15 15
[f]´([psi]´)= ---Ua^(-2), [f]([psi]´) = ---U[psi]´a^(-2); (15)
2 2

and (10) reduces to
_ _
p 9 | / x² \² / y² \² |
----- + V - ---U | ( --- -1 ) - ( --- - ½ ) | = constant; (16)
[rho] 8 |_ \ a² / \ a² / _|

this gives the state of motion in M. J. M. Hill's spherical vortex,
advancing through the surrounding liquid with uniform velocity.

39. As an application of moving axes, consider the motion of liquid
filling the ellipsoidal case

x² y² z²
--- + --- + --- = 1; (1)
a² b² c²

and first suppose the liquid to be frozen, and the ellipsoid to be
rotating about the centre with components of angular velocity [xi],
[eta], [zeta]; then

u = - y[zeta] + z[eta], v = - z[xi] + x[zeta],
w = - x[eta] + y[xi]. (2)

Now suppose the liquid to be melted, and additional components of
angular velocity [Omega]1, [Omega]2, [Omega]3 communicated to the
ellipsoidal case; the additional velocity communicated to the liquid
will be due to a velocity-function

b² - c² c² - a² a² - b²
[phi] = - [Omega]1 ------- yz - [Omega]2 -------zx - [Omega]3 -------xy, (3)
b² + c² c² + a² a² + b²

as may be verified by considering one term at a time.

If u´, v´, w´ denote the components of the velocity of the liquid
relative to the axes,

2a² 2a²
u´ = u + yR - zQ = ------- [Omega]3 y - ------- [Omega]2 z, (4)
a² + b² c² + a²

2b² 2b²
v´ = v + zP - xR = ------- [Omega]1 z - ------- [Omega]3 x, (5)
b² + c² a² + b²

2c² 2c²
w´ = w + xQ - yP = ------- [Omega]2 x - ------- [Omega]1 y, (6)
c² + a² b² + c²

P = [Omega]1 + [xi], Q = [Omega]2 + [eta], R = [Omega]3 + [zeta]. (7)

Thus

x y z
u´ --- + v´ --- + w´ --- = 0, (8)
a2 b2 c2

so that a liquid particle remains always on a similar ellipsoid.

The hydrodynamical equations with moving axes, taking into account the
mutual gravitation of the liquid, become

1 dp du du du du
----- -- + 4[pi][rho]Ax + -- - vR + wQ + u´-- + v´-- + w´-- = 0, ... , ... , (9)
[rho] dx dt dx dy dz

where
_
/ [oo] abcd[lambda]
A, B, C = | ----------------------------------------------
_/ 0 (a² + [lambda], b² + [lambda], c² + [lambda])P

P² = 4(a² + [lambda]) (b² + [lambda]) (c² + [lambda]). (10)

With the values above of u, v, w, u´, v´, w´, the equations become of
the form

1 dp
----- -- + 4[pi][rho]Ax + [alpha]x + hy + gz = 0, (11)
[rho] dx

1 dp
----- -- + 4[pi][rho]By + hx + [beta]y + fz = 0, (12)
[rho] dy

1 dp
----- -- + 4[pi][rho]Cz + gx + fy + [gamma]z = 0, (13)
[rho] dz

and integrating

p[rho]^(-1) + 2[pi][rho](Ax² + By² + Cz²)

+ ½([alpha]x² + [beta]y² + [gamma]z² + 2fyz + 2gzx + 2hxy) = const., (14)

so that the surfaces of equal pressure are similar quadric surfaces,
which, symmetry and dynamical considerations show, must be coaxial
surfaces; and f, g, h vanish, as follows also by algebraical
reduction; and

4c²(c² - a²) / c² - a² \²
[alpha] = ------------[Omega]2² - ( -------[Omega]2 - [eta] )
(c² + a²)² \ c² + a² /

4b²(a² - b²) / a² - b² \²
- ------------[Omega]3² - ( -------[Omega]3 - [zeta] ), (15)
(a² + b²)² \ a² + b² /

with similar equations for [beta] and [gamma].

If we can make

(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].

In a state of steady motion

d[zeta] [Omega]1 [Omega]2
------- = 0, -------- = --------, (27)
dt [xi] [eta]

[phi] = [psi] = nt, suppose, (28)

[Omega]1[xi] + [Omega]2 [eta] = [Omega][omega], (29)

d[phi] a² + c² [omega]
------ = [zeta]- ------- -------[zeta], (30)
dt a² - c² [Omega]

d[psi] 2a² [Omega]
------ = - ------- -------[zeta], (31)
dt a² + c² [omega]

a² + c² [omega] 2a² [Omega]
1 - ------- ------- = - ------- -------, (32)
a² - c² [Omega] a² + c² [omega]

/ [omega] a² + c² \² (a² - c²)(9a² - c²)
( ------- - ½ ------- ) = -------------------, (33)
\ [Omega] a² - c² / 4(a² + c²)

and a state of steady motion is impossible when 3a > c > a.

An experiment was devised by Lord Kelvin for demonstrating this, in which the difference of steadiness was shown of a copper shell filled with liquid and spun gyroscopically, according as the shell was slightly oblate or prolate. According to the theory above the stability is regained when the length is more than three diameters, so that a modern projectile with a cavity more than three diameters long should fly steadily when filled with water; while the old-fashioned type, not so elongated, would be highly unsteady; and for the same reason the gas bags of a dirigible balloon should be over rather than under three diameters long.

40. _A Liquid Jet._--By the use of the complex variable and its conjugate functions, an attempt can be made to give a mathematical interpretation of problems such as the efflux of water in a jet or of smoke from a chimney, the discharge through a weir, the flow of water through the piers of a bridge, or past the side of a ship, the wind blowing on a sail or aeroplane, or against a wall, or impinging jets of gas or water; cases where a surface of discontinuity is observable, more or less distinct, which separates the running stream from the dead water or air.

Uniplanar motion alone is so far amenable to analysis; the velocity
function [phi] and stream function [psi] are given as conjugate
functions of the coordinates x, y by

w = [f](z) where z = x + yi, w = [phi] + [psi]i, (1)

and then

dw d[phi] d[psi]
-- = ------ + i------ = -u + vi; (2)
dz dx dx

so that, with u = q cos [theta], v = q sin [theta], the function

dz Q Q Q
[zeta] = -Q -- = -------- = ---(u + vi) = --- (cos [theta] + i sin [theta]), (3)
dw (u - vi) q² q

gives [zeta] as a vector representing the reciprocal of the velocity q
in direction and magnitude, in terms of some standard velocity Q.

To determine the motion of a jet which issues from a vessel with plane
walls, the vector [zeta] must be Constructed so as to have a constant
direction [theta] along a plane boundary, and to give a constant skin
velocity over the surface of a jet, where the pressure is constant.

It is convenient to introduce the function

[Omega] = log [zeta] = log(Q/q) + [theta]i (4)

so that the polygon representing [Omega] conformally has a boundary
given by straight lines parallel to the coordinate axes; and then to
determine [Omega] and w as functions of a variable u (not to be
confused with the velocity component of q), such that in the conformal
representation the boundary of the [Omega] and w polygon is made to
coincide with the real axis of u.

It will be sufficient to give a few illustrations.

Consider the motion where the liquid is coming from an infinite
distance between two parallel walls at a distance xx´ (fig. 4), and
issues in a jet between two edges A and A´; the wall xA being bent at
a corner B, with the external angle [beta] = ½[pi]/n.

The theory of conformal representation shows that the motion is given
by
_ _
| [root](b - a´·u - a) + [root](b - a·u - a´) |^1/n
[zeta] = | ------------------------------------------- | , u = ae^(-[pi]w/m); (5)
|_ [root](a - a´·u - b) _|

where u = a, a´ at the edge A, A¹; u = b at a corner B; u = 0 across
xx´ where [phi] = [oo]; and u = [oo], [phi] = [oo] across the end JJ´
of the jet, bounded by the curved lines APJ, A´P´J´, over which the
skin velocity is Q. The stream lines xBAJ, xA´J´ are given by [psi] =
0, m; so that if c denotes the ultimate breadth JJ´ of the jet, where
the velocity may be supposed uniform and equal to the skin velocity Q,

m = Qc, c = m/Q.

If there are more B corners than one, either on xA or x´A´, the
expression for [zeta] is the product of corresponding factors, such as
in (5).

Restricting the attention to a single corner B,

/ Q \^n [root](b - a´·u - a) + [root](b - a·u - a´)
[zeta]^n = ( --- ) (cos n[theta] + i sin n[theta] = -------------------------------------------, (6)
\ q / [root](a - a´.u - b)

/ Q \^n / Q \^n
ch n[omega] = ch log( --- ) cos n[theta] + i sh log ( --- ) sin n[theta]
\ q / \ q /

/b - a´ /u - a´
= ½([zeta]^n + [zeta]^(-n)) = / ------ / ------ (7)
\/ a - a´ \/ u - b´

/ Q \ / Q \^n
sh n[Omega] = sh log ( --- ) cos n[theta] + i ch log ( --- ) sin n[theta]
\ q / \ q /

/b - a´ /u - a´
= ½([zeta]^n - [zeta]^(-n)) = / ------ / ------ (8)
\/ a - a´ \/ u - b´

[oo] > a > b > 0 > a´ > -[oo] (9)

and then

d[Omega] 1 [root](b - a´·b - a´) dw m
-------- = - -- ---------------------------, -- = - ------ (10)
du 2n (u - b)[root](a - a·u - a´) du [pi]u´

the formulas by which the conformal representation is obtained.

For the [Omega] polygon has a right angle at u = a, a´, and a zero
angle at u = b, where [theta] changes from 0 to ½[pi]/n and [Omega]
increases by ½i[pi]/n; so that

d[Omega] A [root](b - a·b - a´)
-------- = ---------------------------, where A = --------------------. (11)
du (u - b)[root](u - a·u - a´) 2n

And the w polygon has a zero angle at u = 0, [oo], where [psi] changes
from 0 to m and back again, so that w changes by im, and

dw B m
-- = ---, where B = - ----. (12)
du u [pi]

Along the stream line xBAPJ,

[psi] = 0, u = ae^(-[pi][phi]/m); (13)

and over the jet surface JPA, where the skin velocity is Q,

d[phi]
------ = -q = -Q, u = ae^([pi]sQ/m) = ae^([pi]s/c), (14)
ds

denoting the arc AP by s, starting at u = a;

/b - a´ /u - a´
ch n[Omega] = cos n[theta] = / ----- / ------ (15)
\/ a - a´ \/ u - b´

/a - b /u - a´
sh n[Omega] = i sin n[theta] = i / ------ / ------ (16)
\/ a - a´ \/ u - b´

[oo] > u = ae^([pi]s/c) > a, (17)

and this gives the intrinsic equation of the jet, and then the radius
of curvature

ds 1 d[phi] i dw i dw / d[Omega]
[rho] = - -------- = --- -------- = --- ------- = --- -- / -------
d[theta] Q d[theta] Q d[Omega] Q du / du

c u - b [root](u - a·u - a´)
= ----·2n----- --------------------, (18)
[pi] u [root](a - b·b - a´)

not requiring the integration of (11) and (12)

If [theta] = [alpha] across the end JJ´ of the jet, where u = [oo], q
= Q,

/b - a´ /a - b
ch n[Omega] = cos n[alpha] = /-------, sh n[Omega] = i sin n[alpha]= / ------, (19)
\/ a - a´ \/ a - a´

Then

a - b·b - a´ a - a´
cos 2n[alpha] - cos 2n[theta] = 2------------ = ½ sin² 2n[alpha]------
a - a´·u - b u - b

[root](a - b.b - a´)[root](u - a·u - b´)
sin 2n[theta] = 2---------------------------------------- (20)
a - a´·u - b

[root](a - a·b - a´)
= sin 2n[alpha]--------------------;
u - b

2n c / b \ [root](a - b·b - a´)
----- ----- = ( 1 + ----- ) -------------------- (21)
[phi] [rho] \ u - b/ [root](u - a·u - a´)

a - a´ + (a + a´) cos 2n[alpha] - [a + a´ + (a - a´) cos 2n[alpha] cos 2n[theta]
= --------------------------------------------------------------------------------
(a - a´) sin² 2n[alpha]

cos 2n[alpha] - cos 2n[theta]
× -----------------------------
sin 2n[theta]

Along the wall AB, cos n[theta] = 0, sin n[theta] = 1,

a > u > b, (22)

/ Q \^n /b - a´ /a - u
ch n[Omega] = i sh log ( --- ) = i / ------ / ------ (23)
\ q / \/ a - a´ \/ u - b´

/ Q \^n /a - b /u - a´
sh n[Omega] = i ch log ( --- ) = i / ------ / ------ (24)
\ q / \/ a - a´ \/ u - b´

ds ds d[phi] m c Q
-- = ------ ------ = ------ = --- -- (25)
du d[phi] dt [pi]qu [pi] qu
_
AB / a Q du
[pi]-- = | --- --
c _/ b q u
_ _ _
/ | [root](a - b)[root](u - a´) + [root](b - a´)[root](a - u) |^1/n du
= | | --------------------------------------------------------- | --. (26)
_/ |_ [root](a - a´)[root](u - b´) _| u

Along the wall Bx, cos n[theta] = 1, sin n[theta] = 0,

b > u > 0 (27)

/ Q \^n /b - a´ /a - u
ch n[Omega] = ch log ( --- ) = / ------ / ------ (28)
\ q / \/ a - a´ \/ b - u´

/ Q \^n /a - b /u - a´
sh n[Omega] = sh log ( --- ) = / ------ / ------. (29)
\ q / \/ a - a´ \/ b - u

At x where [phi] = [oo], u = 0, and q = q0,

/ Q \^n /b - a´ / a /a - b / -a´
( --- ) = / ------ / --- + / ------ / ---. (30)
\ q0 / \/ a - a´ \/ b \/ a - a´ \/ q

In crossing to the line of flow x´A´P´J´, [psi] changes from 0 to m,
so that with q = Q across JJ´, while across xx´ the velocity is q0, so
that

m = q0.xx´ = Q.JJ´ (31)
_ _
JJ´ q0 | /b - a´ / a /a - b / -a´ |^1/n
-- = --- = | / ------ / --- - / ------ / --- | , (32)
xx´ Q |_ \/ a - a´ \/ b \/ a - a´ \/ q _|

giving the contraction of the jet compared with the initial breadth of
the stream.

Along the line of flow x´A´P´J´, [psi] = m, u = a´e^(-[pi][phi]/m),
and from x´ to A´, cos n[theta] = 1, sin n[theta] = 0,

/ Q \^n /b - a´ /a - u
ch n[Omega] = ch log ( --- ) = / ------ / ------, (33)
\ q / \/ a - a´ \/ b - u´

/ Q \^n /a - b /u - a´
sh n[Omega] = sh log ( --- ) = / ------ / ------. (34)
\ q / \/ a - a´ \/ b - u´

0 > u > a´. (35)

Along the jet surface A´J´, q = Q,

/b - a´ /a - u
ch n[Omega] = cos n[theta] = / ------ / ------, (36)
\/ a - a´ \/ b - u´

/a - b /a´ - u
sh n[Omega] = i sin n[theta] = i / ------ / ------, (37)
\/ a - a´ \/ b - u´

a´ > u = a´e^([pi]/sc) > -[oo], (38)

giving the intrinsic equation.

41. The first problem of this kind, worked out by H. v. Helmholtz, of
the efflux of a jet between two edges A and A1 in an infinite wall, is
obtained by the symmetrical duplication of the above, with n = 1, b =
0, a´ = -[oo], as in fig. 5,

/u - a / -a
ch [Omega] = / -----, sh [Omega] = / ---; (1)
\/ u \/ u

and along the jet APJ, [oo] > u = ae^([pi]s/c) > a,

sh [Omega] = i sin [theta] - i[root](a/u) = ie^(-½[pi]s/c), (2)
_ _
/ [oo] / c c
PM = | sin [theta] ds = | e^(-½[pi]s/c) ds = ----- e^(-½[pi]s/c) = ----- sin [theta], (3)
_/ s _/ ½[pi] ½[pi]

so that PT = c/½[pi], and the curve AP is the tractrix; and the
coefficient of contraction, or

breadth of the jet [pi]
---------------------- = --------. (4)
breadth of the orifice [pi] + 2

A change of [Omega] and [theta] into n[Omega] and n[theta] will give
the solution for two walls converging symmetrically to the orifice AA1
at an angle [pi]/n. With n = ½, the reentrant walls are given of
Borda's mouthpiece, and the coefficient of contraction becomes ½.
Generally, by making a´ = - [oo], the line x´A´ may be taken as a
straight stream line of infinite length, forming an axis of symmetry;
and then by duplication the result can be obtained, with assigned n,
a, and b, of the efflux from a symmetrical converging mouthpiece, or
of the flow of water through the arches of a bridge, with wedge-shaped
piers to divide the stream.

42. Other arrangements of the constants n, a, b, a´ will give the
results of special problems considered by J. M. Michell, _Phil.
Trans._ 1890.

Thus with a´ = 0, a stream is split symmetrically by a wedge of angle
[pi]/n as in Bobyleff's problem; and, by making a = [oo], the wedge
extends to infinity; then

/ b / n
ch n[Omega] = / -----, sh n[Omega] = / -----. (1)
\/ b - u \/ b - u

Over the jet surface [psi] = m, q = Q,

u = - e^([pi][phi]/m) = - be^([pi]²/c),

/ 1 / e^([pi]²/c)
ch [Omega] = cos n[theta] = / ---------------, sh [Omega] = i sin n[theta] = i / ---------------, (2)
\/ e^([pi]²/c) + 1 \/ e^([pi]²/c) + 1

½[pi] ds 2n
e^(½[pi]²/c) = tan n[theta], ----- -------- = -------------. (3)
c d[theta] sin 2n[theta]

For a jet impinging normally on an infinite plane, as in fig. 6, n = 1,

e^(½[pi]²/c) = tan [theta], ch (½[pi]s/c) sin 2[theta] = 1, (4)

sh ½[pi]x/c = cot [theta], sh ½[pi]y/c = tan [theta],

sh ½[pi]x/c sh ½[pi]y/c = 1, e^(½[pi](x + y)/c) = e^(½[pi]x/c) + e^(½[pi]y/c) + 1. (5)

With n = ½, the jet is reversed in direction, and the profile is the
catenary of equal strength.

In Bobyleff's problem of the wedge of finite breadth,

/ b /u - a /b - a / u
ch n[Omega] = / --- / -----, sh n[Omega] = / ----- / -----, (6)
\/ a \/ u - b \/ a \/ u - b

/ b /a - b
cos n[alpha] = / ---, sin n[alpha] = / -----, (7)
\/ a \/ a

and along the free surface APJ, q = Q, [psi] = 0, u = e^(-[pi][phi]/m) = ae^([pi]s/c),

/ e^([pi]²/c) - 1
cos n[theta] = cos n[alpha] / ---------------------------,
\/ e^([pi]²/c) - cos² n[alpha]

cos² n[alpha] sin² n[theta]
e^([pi]²/c) = -----------------------------, (8)
sin² n[theta] - sin² n[alpha]

the intrinsic equation, the other free surface A´P´J´ being given by

cos² n[alpha] sin² n[theta]
e^([pi]²/c) = -----------------------------. (9)
sin² n[alpha] - sin² n[theta]

Putting n = 1 gives the case of a stream of finite breadth disturbed
by a transverse plane, a particular case of Fig. 7.

When a = b, [alpha] = 0, and the stream is very broad compared with
the wedge or lamina; so, putting w = w´(a - b)/a in the penultimate
case, and

u = ae^(-w) [asympt] a - (a - b)w´, (10)

/w´ + 1 / 1
ch n[Omega] = / ------, sh n[Omega] = / --------, (11)
\/ w´ \/ [root]w´

in which we may write

w´ = [phi] + [psi]i. (12)

Along the stream line xABPJ, [psi] = 0; and along the jet surface APJ,
-1 > [phi] > -[oo]; and putting [phi] = -[pi]s/c - 1, the intrinsic
equation is

[pi]s/c = cot² n[theta], (13)

which for n = 1 is the evolute of a catenary.

43. When the barrier AA´ is held oblique to the current, the stream
line xB is curved to the branch point B on AA´ (fig. 7), and so must
be excluded from the boundary of u; the conformal representation is
made now with

d[Omega] [root](b - a·b - a´)
-------- = - ---------------------------- (1)
du (u - b) [root](u - a·u - a´)

dw m 1 m´ 1 m + m´ u - b
-- = - ---- ----- - ---- -----, = - ------ · -------------,
du [pi] u - j [pi] u - j [pi] u - j·u - j´

mj´ + m´j
b = ---------, (2)
m + m´

taking u = [oo] at the source where [phi] = [oo], u = b at the branch
point B, u = j, j´ at the end of the two diverging streams where [phi]
= -[oo]; while [psi] = 0 along the stream line which divides at B and
passes through A, A´; and [psi] = m, -m´ along the outside boundaries,
so that m/Q, m´/Q is the final breadth of the jets, and (m + m´)/Q is
the initial breadth, c1 of the impinging stream. Then

/b - a´ /u - a /b - a /u - a´
ch ½[Omega] = / ------ / -----, sh ½[Omega] = / ------ / ------, (3)
\/ a - a´ \/ u - b \/ a - a´ \/ u - b

2b - a - a´ N
ch [Omega] = ----------- - -----,
a - a´ u - b

/ [root](2·a - u·u - a´)
sh [Omega] = / N----------------------,
\/ u - b

a - b·b - a´
N = 2------------. (4)
a - a´

Along a jet surface, q = Q, and

ch[Omega] = cos [theta] = cos [alpha] - ½sin² [alpha](a - a´)/(u - b), (5)

if [theta] = [alpha] at the source x of the jet xB, where u = [oo];
and supposing [theta] = [beta], [beta]´ at the end of the streams
where u = j, j´,

u - b ½ sin² [alpha] u - j cos[theta] - cos[beta]
----- = -------------------------, ------ = ½ sin² [alpha]-----------------------------------------------------,
a - a´ cos [alpha] - cos [theta] a - a´ (cos [alpha] - cos [beta])(cos [alpha] - cos [theta])

u - j´ cos [theta] - cos [beta]´
----- = ½ sin² [alpha]------------------------------------------------------; (6)
a - a´ (cos [alpha] - cos [beta]´)(cos [alpha] - cos [theta])

and [psi] being constant along a stream line

d[phi] dw ds d[phi] dw du
------ = --, Q -------- = -------- = -- --------,
du du d[theta] d[theta] du d[theta]

[pi]Q ds [pi] ds (cos [alpha] - cos [beta])(cos [alpha] - cos [beta]´)sin[theta]
------ -------- = ---- -------- = ---------------------------------------------------------------------------------,
m + m´ d[theta] c d[theta] (cos [alpha] - cos [theta])(cos [theta] - cos [beta])(cos [theta] - cos [alpha]´)

sin [theta] cos [alpha] - cos [beta]´ sin [theta]
= ------------------------- + ------------------------- · ------------------------
cos [alpha] - cos [theta] cos [beta] - cos [beta]´ cos [theta] - cos [beta]

cos [alpha] - cos [beta] sin [theta]
------------------------ · -------------------------, (7)
cos [beta] - cos [beta]´ cos [theta] - cos [beta]´

giving the intrinsic, equation of the surface of a jet, with proper
attention to the sign.

From A to B, a > u > b, [theta] = 0,

Q a - a´
ch [Omega] = ch log --- = cos [alpha] - ½ sin² [alpha] ------
q a - b

Q [root](a - u·u - a´)
sh [Omega] = sh log --- = -------------------- sin [alpha]
q u - b

Q (u - b) cos [alpha] - ½(a - a´) sin² [alpha] + [root](a - u·u - a´)sin[alpha]
--- = ----------------------------------------------------------------------------- (8)
q u - b

ds ds d[phi] Q dw
Q -- = Q ------ ------ = - --- --
du d[phi] du q du

m + m´ (u - b) cos [alpha] - ½(a - a´) sin² [alpha] + [root](a - u·u - a´) sin [alpha]
= ------ · ------------------------------------------------------------------------------- (9)
[pi] j - u·u - j´
_
AB / a (2b - a - a´)(u - b) - 2(a - b)(b - a´) + 2[root](a - b·b -a´·a - u·u -a´)
[pi]-- = | -------------------------------------------------------------------------- du, (10)
c _/ b a - a´·j - u·u - j´

with a similar expression for BA´.

The motion of a jet impinging on an infinite barrier is obtained by
putting j = a, j´ = a´; duplicated on the other side of the barrier,
the motion reversed will represent the direct collision of two jets of
unequal breadth and equal velocity. When the barrier is small compared
with the jet, [alpha] = [beta] = [beta]´, and G. Kirchhoff's solution
is obtained of a barrier placed obliquely in an infinite stream.

Two corners B1 and B2 in the wall xA, with a´ = -[oo], and n = 1, will
give the solution, by duplication, of a jet issuing by a reentrant
mouthpiece placed symmetrically in the end wall of the channel; or
else of the channel blocked partially by a diaphragm across the
middle, with edges turned back symmetrically, problems discussed by J.
H. Michell, A. E. H. Love and M. Réthy.

When the polygon is closed by the walls joining, instead of reaching
back to infinity at xx´, the liquid motion must be due to a source,
and this modification has been worked out by B. Hopkinson in the
_Proc. Lond. Math. Soc._, 1898.

Michell has discussed also the hollow vortex stationary inside a
polygon (_Phil. Trans._, 1890); the solution is given by

ch n[Omega] = sn w, sh n[Omega] = i cn w (11)

so that, round the boundary of the polygon, [psi] = K´, sin n[theta] =
0; and on the surface of the vortex [psi] = 0, q = Q, and

cos n[theta] = sn [phi], n[theta] = ½[pi] - am s/c, (12)

the intrinsic equation of the curve.

This is a closed Sumner line for n = 1, when the boundary consists of
two parallel walls; and n = ½ gives an Elastica.

44. _The Motion of a Solid through a Liquid._--An important problem in
the motion of a liquid is the determination of the state of velocity
set up by the passage of a solid through it; and thence of the
pressure and reaction of the liquid on the surface of the solid, by
which its motion is influenced when it is free.

Beginning with a single body in liquid extending to infinity, and
denoting by U, V, W, P, Q, R the components of linear and angular
velocity with respect to axes fixed in the body, the velocity function
takes the form

[phi] = U_[phi]1 + V_[phi]2 + W_[phi]3 + P_[chi]1 + Q_[chi]2 + R_[chi]3, (1)

where the [phi]'s and [chi]'s are functions of x, y, z, depending on
the shape of the body; interpreted dynamically, C - [rho][phi]
represents the impulsive pressure required to stop the motion, or C +
[rho][phi] to start it again from rest.

The terms of [phi] may be determined one at a time, and this problem
is purely kinematical; thus to determine [phi]1, the component U alone
is taken to exist, and then l, m, n, denoting the direction cosines of
the normal of the surface drawn into the exterior liquid, the function
[phi]1 must be determined to satisfy the conditions

(i.) [nabla]²[phi]1 = 0. throughout the liquid;

(ii.) d[ph]1/d[upsilon] = -l, the gradient of [phi] down the normal at
the surface of the moving solid;

(iii.) d[ph]1/d[upsilon] = 0, over a fixed boundary, or at infinity;

similarly for [phi]2 and [phi]3.

To determine [chi]1 the angular velocity P alone is introduced, and
the conditions to be satisfied are

(i.) [nabla]²[chi]1 = 0, throughout the liquid;

(ii.) d[chi]1/d[upsilon] = mz - ny, at the surface of the moving body,
but zero over a fixed surface, and at infinity; the same for [chi]2
and [chi]3.

For a cavity filled with liquid in the interior of the body, since the
liquid inside moves bodily for a motion of translation only,

[phi]1 = -x, [phi]2 = -y, [phi]3 = -z; (2)

but a rotation will stir up the liquid in the cavity, so that the
[chi]'s depend on the shape of the surface.

The ellipsoid was the shape first worked out, by George Green, in his
_Research on the Vibration of a Pendulum in a Fluid Medium_ (1833);
the extension to any other surface will form an important step in this
subject.

A system of confocal ellipsoids is taken

x² y² z²
------------- + ------------- + ------------- = 1 (3)
a² + [lambda] b² + [lambda] c² + [lambda]

and a velocity function of the form

[phi] = x[psi], (4)

where [psi] is a function of [lambda] only, so that [psi] is constant
over an ellipsoid; and we seek to determine the motion set up, and the
form of [psi] which will satisfy the equation of continuity.

Over the ellipsoid, p denoting the length of the perpendicular from
the centre on a tangent plane,

px py pz
l = -------------, m = -------------, n = ------------- (5)
a² + [lambda] b² + [lambda] c² + [lambda]

p²x² p²y² p²z²
1 = ---------------- + ---------------- + ----------------, (6)
(a² + [lambda])² (b² + [lambda])² (c² + [lambda])²

p² = (a² + [lambda])l² + (b² + [lambda])m² + (c² + [lambda])n², (7)
= a²l² + b²m² + c²n² + [lambda],

dp d[lambda]
2p-- = ---------; (8)
ds ds

Thence

d[phi] dx d[psi]
------ = --[psi] + x------
ds ds ds

dx d[psi] dp
= --[psi] + 2(a² + [lambda])--------- l--, (9)
ds d[lambda] ds

so that the velocity of the liquid may be resolved into a component
-[psi] parallel to Ox, and -2(a² + [lambda])l d[psi]/d[lambda] along
the normal of the ellipsoid; and the liquid flows over an ellipsoid
along a line of slope with respect to Ox, treated as the vertical.

Along the normal itself

d[phi] / d[psi] \
----- = ( [psi] + 2(a² + [lambda])-------- )l, (10)
ds \ d[lambda] /

so that over the surface of an ellipsoid where [lambda] and [psi] are
constant, the normal velocity is the same as that of the ellipsoid
itself, moving as a solid with velocity parallel to Ox

d[psi]
U = -[psi] - 2(a² + [lambda])---------, (11)
d[lambda]

and so the boundary condition is satisfied; moreover, any ellipsoidal
surface [lambda] may be supposed moving as if rigid with the velocity
in (11), without disturbing the liquid motion for the moment.

The continuity is secured if the liquid between two ellipsoids
[lambda] and [lambda]1, moving with the velocity U and U1 of equation
(11), is squeezed out or sucked in across the plane x = 0 at a rate
equal to the integral flow of the velocity [psi] across the annular
area [alpha]1 - [alpha] of the two ellipsoids made by x = 0; or if
_
/ [lambda]1 d[alpha]
[alpha]U - [alpha]1U1 = | [psi]-------- d[lambda], (12)
_/ [lambda] d[lambda]

[alpha] = [pi][root](b² + [lambda]·c² + [lambda]). (13)

d[psi] d / d[psi] \
3[alpha]--------- + 2(a² + [lambda])-------- ( [alpha]-------- ) = 0, (15)
d[lambda] d[lambda] \ d[lambda]/

and integrating

d[psi]
(a² + [lambda])^3/2 [alpha]-------- = a constant, (16)
d[lambda]

so that we may put
_
/ Md[lambda]
[psi] = | ----------------, (17)
_/ (a² + [lambda])P

P² = 4(a² + [lambda])(b² + [lambda])(c² + [lambda]), (18)

where M denotes a constant; so that [psi] is an elliptic integral of
the second kind.

The quiescent ellipsoidal surface, over which the motion is entirely
tangential, is the one for which

d[psi]
2(a² + [lambda])--------- + [psi] = 0, (19)
d[lambda]

and this is the infinite boundary ellipsoid if we make the upper limit
[lambda]1 = [oo].

The velocity of the ellipsoid defined by [lambda] = 0 is then

d[psi]0
U = -2a²--------- - [psi]0
d[lambda]
_
M / [oo] Md[lambda]
= --- - | ----------------
abc _/ 0 (a² + [lambda])P

M
= --- (1 - A0), (20)
abc

with the notation
_
/ [oo] abc d[lambda]
A or A_[lambda] = | ----------------
_/[lambda] (a² + [lambda])P
_
d / [oo] d[lambda]
= -2abc--- | ---------, (21)
da²_/ [lambda] P

so that in (4)

M UxA xA_[lambda]
[phi] = ---xA = ------, [phi]1 = -----------, (22)
abc 1 - A0 1 - A0

in (1) for an ellipsoid.

The impulse required to set up the motion in liquid of density [rho]
is the resultant of an impulsive pressure [rho][phi] over the surface
S of the ellipsoid, and is therefore
_ _ _ _
/ / / /
| | [rho][phi]l dS = [rho][psi]0 | | xl dS
_/_/ _/_/

= [rho][psi]0 (volume of the ellipsoid) = [psi]0 W´, (23)

where W´ denotes the weight of liquid displaced.

Denoting the effective inertia of the liquid parallel to Ox by
[alpha]W´. the momentum

[alpha]W´U = [psi]0W´ (24)

[psi]0 A0
[alpha] = ------ = ------; (25)
U 1 - A0

in this way the air drag was calculated by Green for an ellipsoidal
pendulum.

Similarly, the inertia parallel to Oy and Oz is

B0 C0
[beta]W´ = ------ W´, [gamma]W´ = ------ W´, (26)
1 - B0 1 - C0
_
/ [oo] abc d[lambda]
B_[lambda], C_[lambda] = | -------------------------------; (27)
_/[lambda] (b² + [lambda], c² + [lambda])P

and

A + B + C = abc/½P, A0 + B0 + C0 = 1. (28)

For a sphere

a = b = c, A0 = B0 = C0 = 1/3, [alpha] = [beta] = [gamma] = ½, (29)

so that the effective inertia of a sphere is increased by half the
weight of liquid displaced; and in frictionless air or liquid the
sphere, of weight W, will describe a parabola with vertical
acceleration

W - W´
------- g. (30)
W + ½W´

Thus a spherical air bubble, in which W/W´ is insensible, will begin
to rise in water with acceleration 2g.

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