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a perfect

component ý or of the velocity be reversed, the kinetic energy Kinetics of
will clearly be unchanged, and hence no terms ÿż, żż, or ỷ can liquid.
appear in the expression for the kinetic energy: which, on this
account, and because of the symmetry of circumstances with
reference to y and 2, is

T = {{Px2 + Q (y2 + ¿3)}.

Also, we see that P and Q are functions of a simply, since the
circumstances are similar for all values of У and z.
Hence, by

differentiation,

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dQ Qÿ +

dx

dT

dz

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and the equations of motion are

*,

dP d/d7
dx dt dy

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(dP dQ
P£ + } {√ - (+)} = X,
x2
dx

Pä 1

+2

da

dQ
yx = Y, Qz+ zx= Z.
dx

Principles sufficient for a practical solution of the problem of
determining P and Q will be given later. In the meantime, it
is obvious that each decreases as x increases.
Hence the equa-

tions of motion show that

tion of a ball

321. A ball projected through a liquid perpendicularly Effect of a from an infinite plane boundary, and influenced by no other on the moforces than those of fluid pressure, experiences a gradual ac- through a liquid. celeration, quickly approximating to a limiting velocity which it sensibly reaches when its distance from the plane is many times its diameter. But if projected parallel to the plane, it experiences, as the resultant of fluid pressure, a resultant attraction towards the plane. The former of these results is easily proved by first considering projection towards the plane (in which case the motion of the ball will obviously be retarded), and by taking into account the general principle of reversibility (§ 272) which has perfect application in the ideal case of a perfect liquid. The second result is less easily foreseen without

Seeming attraction between two ships

the aid of Lagrange's analysis; but it is an obvious consequence of the Hamiltonian form of his equations, as stated in words in § 319 above. In the precisely equivalent case, of a liquid extending infinitely in all directions, and given at rest; moving side and two equal balls projected through it with equal velocities

by side in the same direction.

perpendicular to the line joining their centres-the result that the two balls will seem to attract one another is most remarkable, and very suggestive.

Hydrodynamical

examples continued.

"Centre of reaction defined.

""

Example (2).—A solid symmetrical round an axis, moving through a liquid so as to keep its axis always in one plane. Let be the angular velocity of the body at any instant about any axis perpendicular to the fixed plane, and let u and q be the component velocities along and perpendicular to the axis of figure, of any chosen point, C, of the body in this line. By the general principle stated in § 320 (since changing the sign of u cannot alter the kinetic energy), we have

T= } (Au3 + Bq3 + μ'w2 + 2Ewq).....

.(a),

where A, B, p', and E are constants depending on the figure of the body, its mass, and the density of the liquid. Now let v denote the velocity, perpendicular to the axis, of a point which we shall call the centre of reaction, being a point in the axis and at a distance from C, so that (§ 87) q = v w. Then,

E

E
B

B

E2 denoting μ'- by μ, we have T = } (Au3 + Bv3 + μw2)....................(a'). B Let x and y be the co-ordinates of the centre of reaction relatively to any fixed rectangular axes in the plane of motion of the axis of figure, and let be the angle between this line and OX, at any instant, so that

@=

= 0, u = x cos 0 + î sin 0, v = − ✯ sin 0 ÷ ý cos 0........................ (b). Substituting in T, differentiating, and retaining the notation. u, v where convenient for brevity, we have

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μ

Hence the equations of motion are
μÖ – (A – B) uv = L,

d(Au cos 0 - Bv sin ()
dt

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where X, Y are the component forces in lines through C parallel to OX and OY, and L the couple, applied to the body.

હૃ

η

Denoting by A, έ, 7 the impulsive couple, and the components of impulsive force through C, required to produce the motion at any instant, we have of course [§ 313 (c)],

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'cos2 0

=

and therefore by (c), and (b),

1

λ

u = A

1 ( cos 0 + n sin 0), v = 1 (− sin 0 + cos 0), 6 =

B

μ

A

dT

do

μ

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=

+

+

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μθ, ξ=

sing (0)

B

d(Au sin + Bv cos 0)

dt

ino 9),

B

dT

dx

sin θ cos θξ +

, ης

S

sin 0 cos On,

+ (1 - 13 )
(sin2

0

=

dT

- {(− §2 + ŋ3) sin 20 + 2έn cos 20} = L,

dt

A - B
2AB

and the first of equations (h) becomes

ᏆᎾ Ꭺ -- Ᏼ
+

dt2

2AB

+

cos 0

and the equations of motion become

d°0 A B 2AB

dt

The simple case of X=0, Y=0, L=0, is particularly interesting.
In it έ and are each constant; and we may therefore choose the
axes OX, OY, so that n shall vanish. Thus we have, in (g), two
first integrals of the equations of motion; and they become

cos2 0
A

sin 20.........

In this let, for a moment, 20 = 4, and

μ +gh W sin &= 0,

ἀφ
dt

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A-B
ᎪᏴ

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B

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n

(d),

(e),

·(ƒ),

· (9),

=X, dn=Y, (h).

dt

· (k):

·(1).

=gh W. It becomes

which is the equation of motion of a common pendulum, of mass W, moment of inertia μ round its fixed axis, and length

Hydrodynamical examples continued.

Hydrodynamical examples continued.

h from axis to centre of gravity; if be the angle from
the position of equilibrium to the position at time t. As we
shall see, under kinetics, the final integral of this equation
expresses in terms of t by means of an elliptic function.
By using the value thus found for or, in (k), we have
equations giving x and y in terms of t by common integration;
and thus the full solution of our present problem is reduced to
quadratures. The detailed working out to exhibit both the actual
curve described by the centre of reaction, and the position of
the axis of the body at any instant, is highly interesting. It is
very easily done approximately for the case of very small angular
vibrations; that is to say, when either A-B is positive, and

always very small, or A-B negative, and very nearly
equal to . But without attending at present to the final
integrals, rigorous or approximate, we see from (k) and (1) that

322. If a solid of revolution in an infinite liquid, be set in motion round any axis perpendicular to its axis of figure, or simply projected in any direction without rotation, it will move with its axis always in one plane, and every point of it moving only parallel to this plane; and the strange evolutions which it will, in general, perform, are perfectly defined by comparison with the common pendulum thus. First, for brevity, we shall Quadrantal call by the name of quadrantal pendulum (which will be further exemplified in various cases described later, under electricity and magnetism; for instance, an elongated mass of soft iron pivoted on a vertical axis, in a "uniform field of magnetic force"), a body moving about an axis, according to the same law with reference to a quadrant on each side of its position of equilibrium, as the common pendulum with reference to a half circle on each side.

defined.

Let now the body in question be set in motion by an impulse, , in any line through the centre of reaction, and an impulsive couple λ in the plane of that line and the axis. This will (as will be proved later in the theory of statical couples) have the same effect as a simple impulse § (applied to a point, if not of the real body, connected with it by an imaginary infinitely light framework) in a certain fixed line, which we shall call the line of resultant impulse, or of resultant momentum,

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being parallel to the former line, and at a distance from it equal to Motion of

a solid of revolution

through a

λ

&

The whole momentum of the motion generated is of course (§ 295) equal to . The body will move ever afterwards according to the following conditions :-(1.) The angular velocity follows the law of the quadrantal pendulum. (2.) The distance of the centre of reaction from the line of resultant impulse varies simply as the angular velocity. (3.) The velocity of the centre of reaction parallel to the line of impulse is found by dividing the excess of the whole constant energy of the motion above the part of it due to the angular velocity round the centre of reaction, by half the momentum. (4.) If A, B, and μ denote constants, depending on the mass of the solid and its distribution, the density of the liquid, and the form and dimensions of the solid, such that Ë Ë × Α' Β’μ are the linear velocities, and the angular velocity, respectively produced by an impulse & along the axis, an impulse § in a line through the centre of reaction perpendicular to the axis, and an impulsive couple X in a plane through the axis; the length of the simple gravitation pendulum, whose motion would keep time with the periodic motion in question, gμAB and, when the angular motion is vibratory, the §1(A – B)' vibrations will, according as A > B, or A < B, be of the axis, or of a line perpendicular to the axis, vibrating on each side of the line of impulse. The angular motion will in fact be vibratory if the distance of the line of resultant impulse from the centre of reaction is anything less than (A ~В) μ cos 21 AB

is

√(A~

where a denotes the inclination of the im

pulse to the initial position of the axis. In this case the path of the centre of reaction will be a sinuous curve symmetrical on the two sides of the line of impulse; every time it cuts this line, the angular motion will reverse, and the maximum inclination will be attained; and every time the centre of reaction is at its greatest distance on either side, the angular velocity will be at its greatest, positive or negative, value, and the linear velocity of

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