With liquid of density ρ, this gives rise to a kinetic reaction to
acceleration *d*U/*dt*, given by

πρb^{2} | a^{2} + b^{2} |
d U |
= | a^{2} + b^{2} |
M′ | d U |
, | |

a^{2} ~ b^{2} | dt |
a^{2} ~ b^{2} | dt |

if M′ denotes the mass of liquid displaced by unit length of the
cylinder *r* = *b*. In particular, when *a* = ∞, the extra inertia is M′.

When the cylinder *r* = *a* is moved with velocity U and *r* = *b* with
velocity U_{1} along O*x*,

φ = U | a^{2} |
( | b^{2} |
+ r ) cos θ − U_{1} | b^{2} |
( r + | a^{2} |
) cos θ, |

b^{2} − a^{2} | r |
b^{2} − a^{2} | r |

ψ = −U | a^{2} |
( | b^{2} |
− r ) sin θ − U_{1} | b^{2} |
( r − | a^{2} |
) sin θ, |

b^{2} − a^{2} | r |
b^{2} − a^{2} | r |

and similarly, with velocity components V and V_{1} along O*y*

φ = V | a^{2} |
( | b^{2} |
+ r ) cos θ − V_{1} | b^{2} |
( r + | a^{2} |
) cos θ, |

b^{2} − a^{2} | r |
b^{2} − a^{2} | r |

ψ = V | a^{2} |
( | b^{2} |
− r ) sin θ + V_{1} | b^{2} |
( r − | a^{2} |
) sin θ, |

b^{2} − a^{2} | r |
b^{2} − a^{2} | r |

and then for the resultant motion

w = (U^{2} + V^{2}) | a^{2} |
z |
+ | a^{2}b^{2} |
U + Vi | ||

b^{2} − a^{2} | U + Vi | b^{2} − a^{2} | z |

−(U_{1}^{2} + V_{1}^{2}) | b^{2} |
z |
− | a^{2}b^{2} |
U_{1} + V_{1}i |
. | ||

b^{2} − a^{2} | U_{1} + V_{1}i |
b^{2} − a^{2} | z |

The resultant impulse of the liquid on the cylinder is given by the
component, over *r* = *a* (§ 36),

X = ∫ ρφ cos θ·adθ = πρa^{2} ( U | b^{2} + a^{2} |
− U_{1} | 2b^{2} |
); |

b^{2} − a^{2} | b^{2} − a^{2} |

and over *r* = *b*

X_{1} = ∫ ρφ cos θ·bdθ = πρb^{2} ( U | 2a^{2} |
− U_{1} | b^{2} + a^{2} |
), |

b^{2} − a^{2} | b^{2} − a^{2} |

and the difference X − X_{1} is the component momentum of the liquid
in the interspace; with similar expressions for Y and Y_{1}.

Then, if the outside cylinder is free to move

X_{1} = 0, | V_{1} |
= | 2a^{2} |
, X = πρa^{2}U | b^{2} − a^{2} |
. |

U | b^{2} + a^{2} |
b^{2} + a^{2} |

But if the outside cylinder is moved with velocity U_{1}, and the
inside cylinder is solid or filled with liquid of density σ,

X = −πρa^{2}U, | U_{1} |
= | 2ρb^{2} |
, |

U | ρ (b^{2} + a^{2}) + σ (b^{2} − a^{2}) |

U − U_{1} |
= | (ρ − σ) (b^{2} − a^{2}) |
, |

U_{1} | ρ (b^{2} + a^{2}) + σ (b^{2} − a^{2}) |

and the inside cylinder starts forward or backward with respect to the outside cylinder, according as ρ > or < σ.

30. The expression for ω in (1) § 29 may be increased by the addition of the term

*im*log

*z*= −

*m*θ +

*im*log

*r*,

representing vortex motion circulating round the annulus of liquid.

Considered by itself, with the cylinders held fixed, the vortex
sets up a circumferential velocity *m*/*r* on a radius *r*, so that the
angular momentum of a circular filament of annular cross section *d*A
is ρ*md*A, and of the whole vortex is ρ*m*π (*b*^{2} − *a*^{2}).

Any circular filament can be started from rest by the application
of a circumferential impulse πρ*mdr* at each end of a diameter; so
that a mechanism attached to the cylinders, which can set up a
uniform distributed impulse πρ*m* across the two parts of a diameter
in the liquid, will generate the vortex motion, and react on the
cylinder with an impulse couple −ρ*m*π*a*^{2} and ρ*m*π*b*^{2}, having resultant
ρ*m*π (*b*^{2} − *a*^{2}), and this couple is infinite when *b* = ∞, as the
angular momentum of the vortex is infinite. Round the cylinder
*r* = *a* held fixed in the U current the liquid streams past with velocity

*q*′ = 2U sin θ +

*m*/

*a*;

and the loss of head due to this increase of velocity from U to *q*′ is

q′^{2} − U^{2} |
= | (2U sin θ + m/a)^{2} − U^{2} |
, |

2g | 2g |

so that cavitation will take place, unless the head at a great distance exceeds this loss.

The resultant hydrostatic thrust across any diametral plane
of the cylinder will be modified, but the only term in the loss
of head which exerts a resultant thrust on the whole cylinder is
2*m*U sin θ/*ga*, and its thrust is 2πρ*m*U absolute units in the direction
Cy, to be counteracted by a support at the centre C; the liquid is
streaming past *r* = a with velocity U reversed, and the cylinder is
surrounded by a vortex. Similarly, the streaming velocity V
reversed will give rise to a thrust 2πρ*m*V in the direction *x*C.

Now if the cylinder is released, and the components U and V are reversed so as to become the velocity of the cylinder with respect to space filled with liquid, and at rest at infinity, the cylinder will experience components of force per unit length

(i.) − 2πρ*m*V, 2πρ*m*U, due to the vortex motion;

(ii.) − πρ*a*^{2} *d* U*dt*, − πρ*a*^{2} *d* V*dt*, due to the kinetic reaction of the liquid;

(iii.) 0, −π(σ − ρ) *a*^{2}*g*, due to gravity,

taking O*y* vertically upward, and denoting the density of the cylinder
by σ; so that the equations of motion are

πσa^{2} | dU |
= − πρa^{2} | dU |
− 2πρmV, |

dt | dt |

πσa^{2} | dV |
= − πρa^{2} | dV |
+ 2πρmV − π (σ − ρ) a^{2}g, |

dt | dt |

or, putting *m* = *a*^{2}ω, so that the vortex velocity is due to an angular
velocity ω at a radius *a*,

*d*U/

*dt*+ 2ρωV = 0,

*d*V/

*dt*− 2ρωU + (σ−ρ)

*g*= 0.

Thus with *g* = 0, the cylinder will describe a circle with angular
velocity 2ρω/(σ + ρ), so that the radius is (σ + ρ) *v*/2ρω, if the velocity
is *v*. With σ = 0, the angular velocity of the cylinder is 2ω; in this
way the velocity may be calculated of the propagation of ripples
and waves on the surface of a vertical whirlpool in a sink.

Restoring σ will make the path of the cylinder a trochoid; and so the swerve can be explained of the ball in tennis, cricket, baseball, or golf.

Another explanation may be given of the sidelong force, arising
from the velocity of liquid past a cylinder, which is encircled by a
vortex. Taking two planes *x* = ± *b*, and considering the increase of
momentum in the liquid between them, due to the entry and exit
of liquid momentum, the increase across *dy* in the direction O*y*,
due to elements at P and P′ at opposite ends of the diameter PP′, is

ρ + ρ = 2ρ |

and with *y* = *b* tan θ, *r* = *b* sec θ, this is

*m*U

*d*θ (1 −

*a*

^{2}

*b*

^{−2}cos 3θ cos θ),

and integrating between the limits θ = ±12π, the resultant, as before,
is 2πρ*m*U.

31. *Example 2.—Confocal Elliptic Cylinders.*—Employ the elliptic
coordinates η, ξ, and ζ = η + ξ*i*, such that

*z*=

*c*ch ζ,

*x*=

*c*ch η cos ξ,

*y*=

*c*sh η sin ζ;

then the curves for which η and ξ are constant are confocal ellipses and hyperbolas, and

J = | d(x, y) |
= c^{2} (ch^{2} η − cos^{2} ξ) |

d(η, ξ) |

*c*

^{2}(ch2η − cos 2ξ) =

*r*

_{1}

*r*

_{2}= OD

^{2},

if OD is the semi-diameter conjugate to OP, and *r*_{1}, *r*_{2} the focal
distances,

*r*

_{1},

*r*

_{2}=

*c*(ch η ± cos ξ);

*r*

^{2}=

*x*

^{2}+

*y*

^{2}=

*c*

^{2}(ch

^{2}η − sin

^{2}ξ)

*c*

^{2}(ch 2η + cos 2ξ).

Consider the streaming motion given by

*w*=

*m*ch (ζ − γ), γ = α + β

*i*,

*m*ch (η − α) cos (ξ − β), ψ =

*m*sh (η − α) sin (ξ − β).

Then ψ = 0 over the ellipse η = α, and the hyperbola ξ = β, so that
these may be taken as fixed boundaries; and ψ is a constant on a C_{4}.

Over any ellipse η, moving with components U and V of velocity,

*y*− V

*x*= [

*m*sh (η − α) cos β + U

*c*sh η ] sin ξ

- [ *m* sh (η − α) sin β + Vc ch η ] cos ξ;

so that ψ′ = 0, if

U = − | m |
sh (η − α) | cos β, V = − | m |
sh (η − α) | sin β, | ||

c | sh η | c | ch η |

having a resultant in the direction PO, where P is the intersection of an ellipse η with the hyperbola β; and with this velocity the ellipse η can be swimming in the liquid, without distortion for an instant.

At infinity

U = − | m |
e^{−a} cos β = − | m |
cos β, |

c | a − b |

V = − | m |
e−a sin β = − | m |
sin β, |

c | a − b |

*a* and *b* denoting the semi-axes of the ellipse α; so that the liquid is
streaming at infinity with velocity Q = *m*/(*a* + *b*) in the direction of
the asymptote of the hyperbola β.

An ellipse interior to η = α will move in a direction opposite to
the exterior current; and when η = 0, U = ∞, but V = (*m*/*c*) sh α sin β.

Negative values of η must be interpreted by a streaming motion on a parallel plane at a level slightly different, as on a double Riemann sheet, the stream passing from one sheet to the other across a cut SS′ joining the foci S, S′. A diagram has been drawn by Col. R. L. Hippisley.