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of E that will be altered is that depending on 2^,171^, because <^i becomes

<^j 4- -p^ Zx on account of the displacement. The variation of actual energy due
ux

to the displacement is therefore

hE=-inCt (2'^w,) dnx (41).

But by equation (12) the work done by the mechanical forces on m^ during
the motion is

hW=t ("^^^dv) Bx (42);

and since our hypothesis is a purely mechanical one, we must have by the
conservation of force,

hE+8W=0 (43);

that is, the loss of energy of the vortices must be made up by work done iu
moving magnets, so that

AnCt (2 ^ m,dv\ Bx + X ("^ m,d v) Sx = 0,

<^=l (^^)^

VOL. L 60

474 ON PHYSICAL LINES OF FORCE.

SO that the energy of the vortices in unit of volume is

^/.(a' + ^ + y) (45);

and that of a vortex whose volume is F is

^^(a^ + /3^ + /)F. (46).

In order to produce or destroy this energy, work must be expended on,
or received from, the vortex, either by the tangential action of the layer of
particles in contact with it, or by change of form in the vortex. We shall first
investigate the tangential action between the vortices and the layer of particles
in contact with them.

Prop. VII. — To find the energy spent upon a vortex in unit of time by
the layer of particles which surrounds it.

Let P, Q, R be the forces acting on unity of the particles in the three
co-ordinate directions, these quantities being functions of a;, y, and z. Since
each particle touches two vortices at the extremities of a diameter, the reaction
of the particle on the vortices will be equally divided, and will be

-iP, -IQ, -iR

on each vortex for unity of the particles; but since the superficial density of
the particles is — (see equation (34)), the forces on unit of surface of a vortex
will be

"■4^^' "4^^' "4^^-
Now let dS be an element of the surface of a vortex. Let the direction-cosines
of the normal be I, m, n. Let the co-ordinates of the element be x, y, z. Let
the component velocities of the surface be u, v, w. Then the work expended on
that element of surface will be

'^=-±(Fu + Qv + Rw)dS (47).

Let us begin with the first term, PudS. P may be written

^dP dP

^^^d^'^^d^y

and u^n^'-my.

J, ^dP ^dP dP ....

^" + ^^ + ^2/ + ^^ (48),

ON PHYSICAL LINES OF FORCE.

475

Remembering that the surface of the vortex is a closed one, so that

XnxdS = XmxdS = %mydS = tmzdS = 0,
and XmydS = tnzdS= F,

we find 2P^S=(f^-^r)F

and the whole work done on the vortex in unit of time will be
dE 1

(49).

^=-iz^(Pu + Qv + Rw)dS

0.1

An

1 f /dQ dRX^^fdR dP\^ (dP dQ\\y

:^Hd^-Wy)^^[dx-^z)^y[d^-dx)j^

47r l*Uz

(50).

Prop. VIII. — To find the relations between the alterations of motion of the
vortices, and the forces P, Q, R which they exert on the layer of particles
between them.

Let V be the volume of a vortex, then by (46) its energy is

1

OTT

and

dE 1 Tr/ ^* . /o^/3_L ^y

(51),
.(52).

(53).

dt ' ^ dt ^ dtj
Comparing this value with that given in equation (50), we find

/dQ dR da\ , ^ /dR dP d^\ ^ fdP dQ dy\ .

This equation being true for all values of a, ^, and y, first let yS and y
vanish, and divide by a. We find

dQ_dR_ da^
dz dy~^ dt

^. ., , dR dP d^

and dP_dQ^ dry

dy dx ^ dt

From these equations we may determine the relation between the alterations
of motion -j- , &c. and the forces exerted on the layers of particles between

60—2

(54).

476 ON PHYSICAL LINES OF FORGE.

the vortices, or, in the language of our hypothesis, the relation between changes
in the state of the magnetic field and the electromotive forces thereby brought
into play.

In a memoir "On the Djoiamical Theory of Diffraction" (Cambridge Philo-
sophical Transactions, Vol. ix. Part 1, section 6), Professor Stokes has given a
method by which we may solve equations (54), and find P, Qy and R in tenns
of the quantities on the right hand of those equations. I have pointed out*
the application of this method to questions in electricity and magnetism.

Let us then find three quantities F, G, H from the equations

dG

dz ~

dH "1
■ dy =^''

dH
dx

dF ^

dF
dy-

dG

(55),

with the conditions '^\Ai^°'^'dy^^^dz^'^)^'^^^ ^^^^'

dF dG dH ^ ,^^.

dx dy dz

Differentiating (55) with respect to t, and comparing with (54), we find

-f . ^=f . -f (-)•

We have thus determined three quantities, F, G, H, from which we can
find P, Q, and R by considering these latter quantities as the rates at which
the former ones vary. In the paper already referred to, I have given reasons
for considering the quantities F, G, H as the resolved parts of that which
Faraday has conjectured to exist, and has called the electrotonic state. In that
paper I have stated the mathematical relations between this electrotonic state
and the lines of magnetic force as expressed in equations (55), and also between
the electrotonic state and electromotive force as expressed in equations (58). We
must now endeavour to interpret them from a mechanical point of view in
connexion with our hypothesis.

* Camhridge Philosophical Transactions, Vol. X. Part i. Art. 3. "On Faraday's Lines of Force,'
pp. 205—209 of this vol.

ON PHYSICAL LINES OF FORCE. 477

We shall in the first place examine the process by which the lines of force
are produced by an electric current.

Let AB, Plate VIII. , p. 488, fig. 2, represent a current of electricity in the
direction from A to B. Let the large spaces above and below AB represent the
vortices, and let the small circles separating the vortices represent the layers of
particles placed between them, which in our hypothesis represent electricity.

Now let an electric current from left to right commence in AB. The
row of vortices gh above AB will be set in motion in the opposite direction
to that of a watch. (We shall call this direction +, and that of a watch -.)
We shall suppose the row of vortices kl still at rest, then the layer of particles
between these rows will be acted on by the row gh on their lower sides, and
will be at rest above. If they are free to move, they will rotate in the
negative direction, and will at the same time move from right to left, or in
the opposite direction from the current, and do form an induced electric current.

If this current is checked by the electrical resistance of the medium, the
rotating particles will act upon the row of vortices Jcl, and make them revolve
in the positive direction till they arrive at such a velocity that the motion of
the particles is reduced to that of rotation, and the induce4 current disappears.
If, now, the primary current AB be stopped, the vortices in the row gh will
be checked, while those of the row kl still continue in rapid motion. The
momentum of the vortices beyond the layer of particles pq will tend to move
them from left to right, that is, in the direction of the primary current; but
if this motion is resisted by the medium, the motion of the vortices beyond pq

It appears therefore that the phenomena of induced currents are part of the
process of communicating the rotatory velocity of the vortices from one part of
the field to another.

As an example of the action of the vortices in producing induced currents,
let us take the following case :— Let B, Plate VIIL, p. 488, fig. 3, be a circular
ring, of uniform section, lapped uniformly with covered wire. It may be shewn
that if an electric current is passed through this wire, a magnet placed within
the coil of wire wiU be strongly affected, but no magnetic effect wUl be produced
on any external point. The effect will be that of a magnet bent round till
its two poles are in contact.

If the coil is properly made, no effect on a magnet placed outside it can

478 ON PHYSICAL LINES OF FORCE.

be discovered, whether the current is kept constant or made to vary in strength ;
but if a conducting wire C be made to embrace the ring any number of times,
an electromotive force will act on that wire whenever the current in the coil is
made to vary ; and if the circuit be closed^ there will be an actual current in
the wire C.

This experiment shews that, in order to produce the electromotive force, it
is not necessary that the conducting wire should be placed in a field of magnetic
force, or that lines of magnetic force should pass through the substance of the
wu'e or near it. All that is required is that lines of force should pass through
the circuit of the conductor, and that these lines of force should vary in quantity
during the experiment.

In this case the vortices, of which we suppose the lines of magnetic force
to consist, are all within the hollow of the ring, and outside the ring all is at
rest. If there is no conducting circuit embracing the ring, then, when the
primary current is made or broken, there is no action outside the ring, except
an instantaneous pressure between the particles and the vortices which they
separate. If there is a continuous conducting circuit embracing the ring, then,
when the primary current is made, there will be a current in the opposite
direction through C; and when it is broken, there will be a current through C
in the same direction as the primary current.

We may now perceive that induced currents are produced when the elec-
tricity yields to the electromotive force, — this force, however, still existing
when the formation of a sensible current is prevented by the resistance of the
circuit.

The electromotive force, of which the components are P, Q, R, arises from
the action between the vortices and the interposed particles, when the velocity
of rotation is altered in any part of the field. It corresponds to the pressure
on the axle of a wheel in a machine when the velocity of the driving wheel
is increased or diminished.

The electrotonic state, whose components are F, G, H, is what the electromotive
force would be if the currents, &c. to which the lines of force are due, instead
of arriving at their actual state by degrees, had started instantaneously from
rest with their actual values. It corresponds to the impulse which would act
on the axle of a wheel in a machine if the actual velocity were suddenly given
to the driving wheel, the machine being previously at rest.

ON PHYSICAL LINES OF FORCE. 479

If the machine were suddenly stopped by stopping the driving wheel, each
wheel would receive an impulse equal and opposite to that which it received
when the machine was set in motion.

This impulse may be calculated for any part of a system of mechanism,
and may be called the reduced momentum of the machine for that point. In
the varied motion of the machine, the actual force on any part arising from
the variation of motion may be found by diiferentiating the reduced momentum
with respect to the time, just as we have found that the electromotive force
may be deduced from the electrotonic state by the same process.

Having found the relation between the velocities of the vortices and the
electromotive forces when the centres of the vortices are at rest, we must
extend our theory to the case of a fluid medium containing vortices, and
subject to all the varieties of fluid motion. If we fix our attention on any
one elementary portion of a fluid, we shall find that it not only travels from
one place to another, but also changes its form and position, so as to be elon-
gated in certain directions and compressed in others, and at the same time (in
the most general case) turned round by a displacement of rotation.

These changes of form and position produce changes in the velocity of the
molecular vortices, which we must now examine.

The alteration of form and position may always be reduced to three simple
extensions or compressions in the direction of three rectangular axes, together
with three angular rotations about any set of three axes. We shall first con-
sider the effect of three simple extensions or compressions.

Prop. IX. — To find the variations of a, yS, y in the parallelepiped .r, y, z
when X becomes x-^-hx; y, y + Sy ; and z, z + Bz; the volume of the figure
remaining the same.

By Prop. II. we find for the work done by the vortices against pressure,

hW=p,B{xyz)-^(a'yzBx-i-p:'zxZy-\-'/x2jSz) (59);

and by Prop. VI. we find for the variation of energy,

BE=-^(aBa + ^h^-{-yBy)xyz (60).

477

480 ON PHYSICAL LINES OF FORCE.

The sum SW+BE must be zero by the conservation of energy, and 8 (xyz) = 0,
since xyz is constant; so that

(Sa-af)+^(s^-^|)+y(Sy-y|) = (61).

In order that this should be true independently of any relations between a, /8,
and y, we must have

Sa = a«|, S^=;8j, Sy = y| (62).

Prop. X. — To find the variations of a, /8, y due to a rotation 0^ about the
axis of X from y to 2;, a rotation O^ about the axis of y from z to x, and a
rotation ^3 about the axis of z from ic to y.

The axis of y8 will move away from the axis of x by an angle \$3 ; so
that /8 resolved in the direction of x changes from to —JSO^.

The axis of y approaches that of x by an angle 6^ ; so that the resolved
part of y in direction x changes from to yd^.

The resolved part of a in the direction of x changes by a quantity depending
on the second power of the rotations, which may be neglected. The variations of
a, )8, y from this cause are therefore

8a = yl9,-M, S^ = a^3-y(9„ hy^^d.-aO, (63).

The most general expressions for the distortion of an element produced by
the displacement of its different parts depend on the nine quantities

d ^ d ^ d ^ d ^ d ^ d ^ d ^ d ^ d ^

tJ""' 3^^^' Tz^"' Tx^J- Ty^y- di^' Tx^' Ty^- Tz^-'

and these may always be expressed in terms of nine other quantities, namely,
three simple extensions or compressions,

Zx Zy hz'
^' Y' ~^
along three axes properly chosen, x\ y\ z', the nine direction-cosines of these
axes with their six connecting equations, which are equivalent to three inde-
pendent quantities, and the three rotations 6^, 0,, 0^ about the axes of x, y, z.

Let the direction-cosines of x' with respect to cc, y, z be /„ mj, n^^ those of
y\ \y 7?ij, Tiy and those of z\ Zj, ma, n, ; then we find

ON PHYSICAL LINES OF FORCE. 481

dx X y z

-J- Bx = I,m, — + km, 4- + ^wi,— - d.

dy ' ' X ' ' y

(C4),

witli similar equations for quantities involving Sy and 8z.

Let a, 13', y be the values of a, ^, y referred to the axes x, y, z; then

a=l,a + mJ3 + n,y^

^' = l,a + mS-^n,y I (65).

y = l,a + m^fi + n{y J
We shaU then have ha = kha +a^ ^-l,^' + ye,-^e, {^(:>),

=i^a'^+ij3'K+W^f+ye.-^d. (67).

By substituting the values of a, /3', y, and comparing with equations (64), we
find

^- = 4^ - ^4'"^^^'^ ^''^

as the variation of a due to the change of form and position of the element.
The variations of ^ and y have similar expressions.

Prop. XI.— To find the electromotive forces in a moving body.

The variation of the velocity of the vortices in a moving element is due to
two causes— the action of the electromotive forces, and the change of form and
position of the element. The whole variation of a is therefore

«"=KS-f)^'^"^^^^^4^^^^^^^ ^''\

But since a is a function of x, y. z and t, the variation of a may be aiso written

^'^=Pj^-py-^TJ'^'i^ (^»'-

Equating the two values of Sa and dividing by ht, and remembering that in the

motion of an incompressible medium

d dx ddy d dz_ /^,x

didt^dy dt^ dzdt~^ ^ ^'

vol. l ^1

482

ON PHYSICAL LINES OF FORCE.

id that in the absence of free magnetism

dx dy dz

•(72).

we find
\/dQ
fi\dz

Putting

dy)

d dx
'^'^dz'dt

d dz d dy ^n d dx
'^dzdt~°'dydi ' '^'dyTt

dy dx da dz da dy d^ dx _^da _
dz dt dz dt dy dt dy dt di

and

l/dG

^'ATz-

_\(d'G
~ fi \dz dt

dH\

da
di-

dy)"

d'H\
dydt)

,.(r.3).

• (74),
..(75).

where F, G, and H are the vahies of the electrotonic components for a fixed
point of space, our equation becomes

dx

dz dG

Q + l^y:J7-H-^7u-

d fry dy r^dx

f) = o (-)•

dy V^" ' '^^ dt '^'^ dt

The expressions for the variations of ^ and y give us two other equations
which may be written down from symmetry. The complete solution of the three
equations is

Q

dz
dt

dx

dF _d^
dt dx

^. dG _d^
di ^'^ dt "^ dt dy

dll_d^
dz

„ ^dx dy

(77).

The first and second terms of each equation indicate the effect of the motion
of any body in the magnetic field, the third term refera to changes in the
electrotonic state produced by alterations of position or intensity of magnets
or currents in the field, and ^ is a function of x, y, z, and t, which is inde-
terminate as far as regards the solution of the original equations, but which
may always be determined in any given case from the circumstances of the
problem. The physical interpretation of ^ is, that it is the clectiic tension at
each point of space.

ON PHYSICAL LINP:S OF FORCE. 483

The physical meaning of the terms in the expression for the electromotive
force depending on the motion of the body, may be made simpler by supposing
the field of magnetic force uniformly magnetized with intensity a in the direction
of the axis of x. Then if /, m, n be the direction-cosines of any portion of a
linear conductor, and S its length, the electromotive force resolved in the direction
of the conductor will be

e = S{Pl + Qm + Rn) (78),

' = ^^^{'''jt-''t) (^^)'

that is, the product of /xa, the quantity of magnetic induction over unit of area

multiplied by Sim y, -" ;7r)» the area swept out by the conductor S in unit of

time, resolved perpendicular to the direction of the magnetic force.

The electromotive force in any part of a conductor due to its motion is
therefore measured by the number of lines of magnetic force which it crosses
in unit of time ; and the total electromotive force in a closed conductor is
measured by the change of the number of lines of force which pass through it ;
and this is true whether the change be produced by the motion of the con-
ductor or by any external cause.

In order to understand the mechanism by which the motion of a conductor
across lines of magnetic force generates an electromotive force in that conductor,
we must remember that in Prop. X. we have proved that the change of form
of a portion of the medium containing vortices produces a change of the velocity
of those vortices ; and in particular that an extension of the medium in the
direction of the axes of the vortices, combined with a contraction in all direc-
tions perpendicular to this, produces an increase of velocity of the vortices ;
while a shortening of the axis and bulging of the sides produces a diminution
of the velocity of the vortices.

This change of the velocity of the vortices arises from the internal effects
of change of form, and is independent of that produced by external electro-
motive forces. If, therefore, the change of velocity be prevented or checked,
electromotive forces will arise, because each vortex will press on the surrounding
particles in the direction in which it tends to alter its motion.

Let A, fig. 4, p. 488, represent the section of a vertical wire moving in the
direction of the arrow from west to east, across a system of lines of magnetic force

61—2

484 ON PHYSICAL LINES OF FORCE.

running north and south. The curved lines in fig. 4 represent the lines of fluid
motion about the wire, the wire being regarded as stationary, and the fluid as
having a motion relative to it. It is evident that, from this figure, we can trace
the variations of form of an element of the fluid, as the form of the element
depends, not on the absolute motion of the whole system, but on the relative
motion of its parts.

In front of the wire, that is, on its east side, it will be seen that as the
wire approaches each portion of the medium, that portion is more and more
compressed in the direction from east to west, and extended in the direction
from north to south ; and since the axes of the vortices lie in the north and
south direction, their velocity will continually tend to increase by Prop. X.,
unless prevented or checked by electromotive forces acting on the circumference
of each vortex.

We shall consider an electromotive force as positive when the vortices tend
to move the interjacent particles upwards perpendicularly to the plane of the
paper.

The vortices appear to revolve as the hands of a watch when we look at
them from south to north ; so that each vortex moves upwards on its west side,
and downwards on its east side. In front of the wire, therefore, where each
vortex is striving to increase its velocity, the electromotive force upwards must
be greater on its west than on Its east side. There will therefore be a con-
tinual increase of upward electromotive force from the remote east, where it is
zero, to the front of the moving wire, where the upward force wiU be strongest.

Behind the wire a difierent action takes place. As the wire moves away
from each successive portion of the medium, that portion is extended from east
to west, and compressed from north to south, so as to tend to diminish the
velocity of the vortices, and therefore to make the upward electromotive force
greater on the east than on the west side of each vortex. The upward electro-
motive force wiU therefore increase continually from the remote west, where it
is zero, to the back of the moving wire, where it will be strongest.

It appears, therefore, that a vertical wire moving eastwards will experience
an electromotive force tending to produce in it an upward current. If there
is no conducting circuit in connexion with the ends of the wire, no current will
be formed, and the magnetic forces wHl not be altered ; but if such a circuit
exists, there will be a current, and the lines of magnetic force and the velocity

ON PHYSICAL LINES OF FORCE. 485

of the vortices will be altered from their state previous to the motion of the
wire. The change in the lines of force is shewn in fig. 5. The vortices in
front of the wire, instead of merely producing pressures, actually increase in
velocity, while those behind have their velocity diminished, and those at the
sides of the wire have the direction of their axes altered; so that the final
effect is to produce a force acting on the wire as a resistance to its motion.
We may now recapitulate the assumptions we have made, and the results we
have obtained.

(1) Magneto-electric phenomena are due to the existence of matter under
certain conditions of motion or of pressure in every part of the magnetic field,
and not to direct action at a distance between the magnets or currents. The
substance producing these effects may be a certain part of ordinary matter, or
it may be an aether associated with matter. Its density is greatest in iron,
and least in diaraagnetic substances ; but it must be in all cases, except that of
iron, very rare, since no other substance has a large ratio of magnetic capacity
to what we call a vacuum.

(2) The condition of any part of the field, through which lines of magnetic
force pass, is one of unequal pressure in different directions, the direction of
the lines of force being that of least pressure, so that the lines of force may
be considered lines of tension.

(3) This inequality of pressure is produced by the existence in the medium
of vortices or eddies, having their axes in the direction of the lines of force,
and having their direction of rotation determined by that of the lines of force.

We have supposed that the direction was that of a watch to a spectator
looking from south to north. We might with equal propriety have chosen the
reverse direction, as far as known facts are concerned, by supposing resinous elec-
tricity instead of vitreous to be positive. The effect of these vortices depends
on their density, and on their velocity at the circumference, and is independent
of their diameter. The density must be proportional to the capacity of the
substance for magnetic induction, that of the vortices in air being 1. The
velocity must be very great, in order to produce so powerful effects in so rare
a medium.

The size of the vortices is indeterminate, but is probably very small as
compared with that of a complete molecule of ordinary matter^''.

* The angular momentum of the system of vortices depends on their average diameter ; so tkat if the
diameter were sensible, we might expect that a magnet would behave as if it contained a revoh-ing bodv

486 O^' PHYSICAL LINES OF FORCE.

(4) The vortices are separated from each other by a single layer of round
particles, so that a system of cells is formed, the partitions being these layers
of particles, and the substance of each cell being capable of rotating as a vortex.

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