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(5) The particles forming the layer are in rolling contact with both the
vortices which they separate, but do not rub against each other. They are
perfectly free to roll between the vortices and so to change their place, provided
they teep within one complete molecule of the substance; but in passing from
one molecule to another they experience resistance, and generate irregular
motions, which constitute heat. These particles, in our theory, play the part of
electricity. Their motion of translation constitutes an electric current, their
rotation serves to transmit the motion of the vortices from one part of the
field to another, and the tangential pressures thus called into play constitute
electromotive force. The conception of a particle having its motion connected
with that of a vortex by perfect rolling contact may appear somewhat awkward.
I do not bring it forward as a mode of connexion existing in nature, or even
as that which I would willingly assent to as an electrical hypothesis. It is,
however, a mode of connexion which is mechanically conceivable, and easily
investigated, and it serves to bring out the actual mechanical connexions
between the known electro-magnetic phenomena; so that I venture to say that
any one who understands the provisional and temporary character of this
hypothesis, will find himself rather helped than hindered by it in his search
after the true interpretation of the phenomena.

The action between the vortices and the layers of particles is in part
tangential; so that if there were any slipping or difierential motion between
the parts in contact, there would be a loss of the energy belonging to the
lines of force, and a gradual transformation of that energy into heat. Now we
know that the hues of force about a magnet are maintained for an indefinite
time without any expenditure of energy; so that we must conclude that
wherever there is tangential action between difierent parts of the medium, there
is no motion of slipping between those parts. We must therefore conceive that
the vortices and particles roll together without shpping ; and that the interior
strata of each vortex receive their proper velocities from the exterior stratum
without slipping, that is, the angular velocity must be the same throughout each
vortex.

within it, -and that the existence of this rotation might be detected by experiments on the free rotation of a
magnet. I have made experiments to investigate this question, but have not yet fully tried the apparatus.



ON PHYSICAL LINES OF FORCE. 487

The only process in which electro- magnetic energy is lost and transforaied
into heat, is in the passage of electricity from one molecule to another. In all
other cases the energy of the vortices can only be diminished when an equivalent
quantity of mechanical work is done by magnetic action.

(6) The effect of an electric current upon the surrounding medium is to
make the vortices in contact with the current revolve so that the parts next
to the current move in the same direction as the current. The parts furthest
from the current will move in the opposite direction ; and if the medium is a
conductor of electricity, so that the particles are free to move in any direction,
thfe particles touching the outside of these vortices will be moved in a direction
contrary to that of the current, so that there will be an induced current in
the opposite direction to the primary one.

If there were no resistance to the motion of the particles, the induced
current would be equal and opposite to the primary one, and would continue
as long as the primary current lasted, so that it would prevent all action of
the primary current at a distance. If there is a resistance to the induced
current, its particles act upon the vortices beyond them, and transmit the motion
of rotation to them, till at last all the vortices in the medium are set in
motion with such velocities of rotation that the particles between them have no
motion except that of rotation, and do not produce currents.

In the transmission of the motion from one vortex to another, there arises u
force between the particles and the vortices, by which the particles are pressed
in one direction and the vortices in the opposite direction. We call the force
actino- on the particles the electromotive force. The reaction on the vortices is
equal and opposite, so that the electromotive force cannot move any part of
the medium as a whole, it can only produce currents. When the primary
current is stopped, the electromotive forces all act in the opposite direction.

(7) When an electric current or a magnet is moved in presence of a
conductor, the velocity of rotation of the vortices in any part of the field is
altered by that motion. The force by which the proper amount of rotation is
transmitted to each vortex, constitutes in this case also an electromotive force,
and, if permitted, will produce currents.

(8) When a conductor is moved in a field of magnetic force, the vortices
in it and in its neighbourhood are moved out of their places, and are changed
in form. The force arising from these changes constitutes the electromotive



488 ON PHYSICAL LINES OF FORCE.

force on a moving conductor, and is found by calculation to correspond with
that determined by experiment.

"We have now shewn in w-hat way electro -magnetic phenomena may be
imitated by an imaginary system of molecular vortices. Those who have been
already inclined to adopt an hypothesis of this kind, will find here the con-
ditions which must be fulfilled in order to give it mathematical coherence, and
a comparison, so far satisfactory, between its necessary results and known facts.
Those who look in a different direction for the explanation of the facts, may
be able to compare this theory with that of the existence of currents flowing
freely through bodies, and with that which supposes electricity to act at a
distance with a force depending on its velocity, and therefore not subject to
the law of conservation of energy.

The facts of electro-magnetism are so complicated and various, that the
explanation of any number of them by several different hypotheses must be
interesting, not only to physicists, but to all who desire to understand how
much evidence the explanation of phenomena lends to the credibility of a theory,
or how far we ought to regard a coincidence in the mathematical expression of
two sets of phenomena as an indication that these phenomena are of the same
kind. We know that partial coincidences of this kind have been discovered ;
and the fact that they are only partial is proved by the divergence of the
laws of the two sets of phenomena in other respects. We may chance to find,
in the higher parts of physics, instances of more complete coincidence, which
may require much investigation to detect their ultimate divergence.



NOTE.

Since the first part of this paper was written, I have seen in Crelle's Journal for 1859,
a paper by Prof. Helmholtz on Fluid Motion, in which he has pointed out that the lines
of fluid motion are arranged according to the game laws as the Hnes of magnetic force, the
path of an electric current corresponding to a line of axes of those particles of the fluid
which are in a state of rotation. This is an additional instance of a physical analogy, the
investigation of which may illustrate both electro-magnetism and hydrodynamics.




^ ^






Fig 5



VOL. L PLATE VIIL



JFi^: 7.



Ti^ S.



XigJD.




Tvg:9. a.



ON PHYSICAL LINES OF FORCE. 489



[From the Philosophical Magazine for January and February, 1802.]
PART III.

THE THEORY OF MOLECULAR VORTICES APPLIED TO STATICAL ELECTRICITY.

In the first part of this paper ^^ I have shewn how the forces acting between
ma^ets, electric currents, and matter capable of magnetic induction may be
accounted for on the hypothesis of the magnetic field being occupied with
innumerable vortices of revolving matter, their axes coinciding with the direction
of the magnetic force at every point of the field.

The centrifugal force of these vortices produces pressures distributed in such
a way that the final efiect is a force identical in direction and magnitude
with that li^ich we observe.

In the second partf I described the mechanism by which these rotations
may be made to coexist, and to be distributed according to the known laws
of magnetic lines of force.

I conceived the rotating matter to be the substance of certain cells, divided
from each other by cell-walls composed of particles which are very small com-
pared with the cells, and that it is by the motions of these particles, and their
tangential action on the substance in the cells, that the rotation is communi-
cated from one cell to another.

I have not attempted to explain this tangential action, but it is necessary

to suppose, in order to account for the transmission of rotation from the exterior

to the interior parts of each cell, that the substance in the cells possesses

elasticity of figure, similar in kind, though different in degree, to that observed

in BoUd bodies. The undulatory theory of light requires us to admit this kind

of elasticity in the luminiferous medium, in order to account for transverse

vibrations. We need not then be surprised if the magneto-electric medium

possesses the same property.

♦ PhiL Mag. March, 1861 [pp. 4.51— 466 of this vol.].

t Phil. Mag. April and May, 1861 [pp. 467—488 of this vol.].

VOL. I. ^-



490 ON PHYSICAL LINES OF FORCE.

According to our theory, the particles which forta the pai-titions between
the cells constitute the matter of electricity. The motion of these particles
constitutes an electric current; the tangential force with which the particles
are pressed by the matter of the cells is electromotive force, and the pressure
of the particles on each other corresponds to the tension or potential of the
electricity.

If we can now explain the condition of a body with respect to the
surrounding medium when it is said to be "charged" with electricity, and
account for the forces acting between electrified bodies, we shall have established
a connexion between all the principal phenomena of electrical science.

We know by experiment that electric tension is the same thing, whether
observed in statical or in current electricity; so that an electromotive force
produced by magnetism may be made to charge a Leyden jar, as Ls done by
the coil machine.

When a difference of tension exists in different parts of any body, the
electricity passes, or tends to pass, from places of greater to places of smaller
tension. If the body is a conductor, an actual passage of electricity takes
place; and if the difference of tensions is kept up, the current continues to
flow with a velocity proportional inversely to the resistance, or directly to the
conductivity of the body.

The electric resistance has a very wide range of values, that of the metals
being the smallest, and that of glass being so great that a charge of electricity
has been preserved'"* in a glass vessel for years without penetrating the thick-
ness of the glass.

Bodies which do not permit a current of electricity to flow through them
are called insulators. But though electricity does not flow through them,
the electrical effects are propagated through them, and the amount of these
effects differs according to the nature of the body; so that equally good insu-
lators may act differently as dielectrics t.

Here then we have two independent qualities of bodies, one by which they
allow of the passage of electricity through them, and the other by which they
allow of electrical action being transmitted through them without any electri-
city being allowed to pass. A conducting body may be compared to a porous
membrane which opposes more or less resistance to the passage of a fluid,

* By Professor W. Thomson. t Faraday, Experimental Researc/tes, Series xi.



ON PHYSICAL LINES OF FORCE. 491

while a dielectric is like an elastic membrane which may be impervious to the
fluid, but transmits the pressure of the fluid on one side to that on the other.

As long as electromotive force acts on a conductor, it produces a current
which, as it meets with resistance, occasions a continual transformation of
electrical energy into heat, which is incapable of being restored again as electri-
cal energy by any reversion of the process.

Electromotive force acting on a dielectric produces a state of polarization
of its parts similar in distribution to the polarity of the particles of iron under
the influence of a magnet*, and, like the magnetic polarization, capable of
being described as a state in which every particle has its poles in opposite
conditions.

In a dielectric under induction, we may conceive that the electricity iri
each molecule is so displaced that one side is rendered positively, and the
other negatively electrical, but that the electricity remains entirely connected
with the molecule, and does not pass from one molecule to another.

The eSect of this action on the whole dielectric mass is to produce a
general displacement of the electricity in a certain direction. This displace-
ment does not amount to a current, because when it has attained a certain
value it remains constant, but it is the commencement of a current, and its
variations constitute currents in the positive or negative direction, according as
the displacement is increasing or diminishing. The amount of the displacement
depends on the nature of the body, and on the electromotive force ; so that
if h is the displacement, R the electromotive force, and E a coefficient
depending on the nature of the dielectric,

R=-iTrE'h',

and if r is the value of the electric current due to displacement,

_dh
''' dt'

These relations are Independent of any theory about the internal mechanism
of dielectrics ; but when we find electromotive force producing electric displace-
ment in a dielectric, and when we find the dielectric recovering from its state
of electric displacement with an equal electromotive force, we cannot hel{)

* See Prof. Mossotti, " Discussione Analitica," Memorie della Soc. Itaiiaiui (Modena), Vol. xxiv.
Part 2, p. 49.

G-2— 2



492 ON PHYSICAL LINES OF FORCE.

regarding the phenomena as those of an elastic body, yielding to a pressure,
and recovering its form when the pressure is removed.

According to our hypothesis, the magnetic medium is divided into cells,
separated by partitions formed of a stratum of particles which play the part
of electricity. When the electric particles are urged in any direction, they will,
by their tangential action on the elastic substance of the cells, distort each
cell, and call into play an equal and opposite force arising from the elasticity
of the cells. When the force is removed, the cells will recover their form,
and the electricity will return to its former position.

In the following investigation I have considered the relation between the
displacement and the force producmg it, on the supposition that the cells are
spherical. The actual form of the cells probably does not differ from that of
a sphere sufficiently to make much difference in the numerical result.

I have deduced from this result the relation between the statical and
dynamical measures of electricity, and have shewn, by a comparison of the
electro- magnetic experiments of MM. Kohlrausch and Weber with the velocity
of light as found by M. Fizeau, that the elasticity of the magnetic medium
in air is the same as that of the luminiferous medium, if these two coex-
istent, coextensive, and equally elastic media are not rather one medium.

It appears also from Prop. XV. that the attraction between two electrified
bodies depends on the value of E\ and that therefore it would be less in
turpentine than in air, if the quantity of electricity in each body remains the
same. If, however, the j^otentials of the two bodies were given, the attraction
between them would vary inversely as E\ and would be greater in turpentine
than in air.

Prop. XII. To find the conditions of equilibrium of an elastic sphere
whose surface is exposed to normal and tangential forces, the tangential forces
being proportional to the sine of the distance from a given point on the sphere.

Let the axis of z be the axis of spherical co-ordinates.

Let ^, -q, C be the displacements of any particle of the sphere in the direc-
tions of X, y, and z.

Let p^, p,jy, 2^zz be the stresses normal to planes perpendicular to the three
axes, and let Py^, 2^zx> X>xy be the stresses of distortion in the planes yz, zx,
and xy.



ON PHYSICAL LINES OF FORCE.



493



Let fJL be the coefficient of cubic elasticity, so that if



(80).



P-^d^Ty^S

Let in be the coefficient of rigidity, so that

^-^»=-(i-|)-^« («^)-

Then we have the following equations of elasticity in an isotropic medium,

(82);



with similar equations in y and z, and also

m /dr] dC



P-=2[d-z+dy)'^

In the case of the sphere, let us assume the radius = a, and

i=exz, y) = ezy, i=f{x' + y')+gz' + d

Then p„ = 2(ix-^m){e-irg)z-{-mez=Pyy'

p,, = 2 (/x - ^m) {e-\-g)z + 2mgz



i'» = 2(« + 2/)2



(83).
(84).

(85).



(86),



The equation of internal equilibrium with respect to z is

d d d ^

dxP- + d^P- + dzP'' = '>

which is satisfied in this case if

m(e + 2/+2^) + 2(^-im)(e+^) = (87).

The tangential stress on the surface of the sphere, whose radius is a at
an angular distance from the axis in plane xz,

T={Pxx -Pzz) sin cos 6 +jp„ (cos' 6 - sin' 6) (88)



ma



2m {e+f-g) a sin ^cos'^- ^ (e + 2/) sin ^.



.(89).



494 ON PHYSICAL LINES OF FORCE,

In order that T may be proportional to sin 6, the first term must vanish, and
therefore

9 = ^+f (90).

r=_!^(e + 2/)sin^ (91).

The normal stress on the surface at any point is

N =2^xx sin'^ ^-^Vyy ^'^^'^ ^ + '^Pxz sin 6 cos d

= 2 (/x - \m) (e+g)a cos 6 + 2ma cos {{e +/) sin"- O + g cos' 6} (92) ;

or by (87) and (90), iV^= -77ia (e + 2/) cos ^ (93).

The tangential displacement of any point is

t = ico9e-Csm0= - {arf+d)sme (94).

The normal displacement is

n = ^sm0 + CGOse = {a'{e+f) + d}cos0 (95).

If we make a'{e+f) + d = (96),

there will be no normal displacement, and the displacement will be entirely
tangential, and we shall have

t = a-esm0 (97).

The whole work done by the superficial forces is

U=^X{Tt)dS,

the summation being extended over the surface of the sphere.

The energy of elasticity in the substance of the sphere is

the summation being extended to the whole contents of the sphere.

We find, as we ought, that these quantities have the same value, namely
U=-^Tra'me{e + 2f) (98).

We may now suppose that the tangential action on the surface arises from a
layer of particles in contact with it, the particles being acted on by their own
mutual pressure, and acting on the surfaces of the two cells with which they
are in contact.



ON rnvsiCAL lines of force. 495

We assume the axis of z to be in the direction of maximum variation of
the pressure among the particles, and we have to determine the relation
between an electromotive force R acting on the particles in that direction, and
the electric displacement h which accompanies it.

Prop. XIII. — To find the relation betNveen electromotive force and electric
displacement when a uniform electromotive force R acts parallel to the axis of z.

Take any element IS of the surface, covered with a stratum whose density
is /3, and having its normal inclined 6 to the axis of 2; then the tangential

force upon it will be

pRhS sin = 2 TBS (99),

T being, as before, the tangential force on each side of the surface. Putting

p = - — as in equation (34)*, we find

R=-27rma(e + 2f) (100).

The displacement of electricity due to the distortion of the sphere is

tSS^ptsmd taken over the whole surface (101)5

and if h is the electric displacement per unit of volume, we shall have

iTTa% = ^i*e (102),

h = ^ae (103);

2tt

so that R = in-m^^-fh (104),

e

or we may write R= — inE-h (105),

e + 2f
provided we assume E'=—7nn~ " (lOG).



or



Finding e and / from (87) and (90), we get



E^ = Trm 1— (107).

3 /x

The ratio of m to /x varies in different substances; but in a medium whose
elasticity depends entirely upon forces acting between pairs of particles, this
ratio is that of G to 5, and in this case

E' = Trm (108).

♦ Phil. Mag. April, ISGl [p. 471 of this vol.].



496 ON PHYSICAL LINES OF FORCE.

When the resistance to compression is infinitely greater than the resistance to
distortion, as in a liquid rendered slightly elastic by gum or jelly,

E'^^TTin (109).

The value of Er must lie between these limits. It is probable that the substance
of our cells is of the former kind, and that we must use the first value of E\
which is that belonging to a hypothetically "perfect" solid■'^ in which

5m = 6/i (110),

so that we must use equation (108).

Prop. XIV. — To correct the equations (9)t of electric currents for the efiect
due to the elasticity of the medium.

We have seen that electromotive force and electric displacement are
connected by equation (105). Differentiating this equation with respect to t, we
find

f= - ^§ (-)■

shewing that when the electromotive force varies, the electric displacement also
varies. But a variation of displacement is equivalent to a current, and this
current must be taken into account in equations (9) and added to r. The three
equations then become

_ J_ /^ _ ^ _ i ^^
-^~ 47r \dv dz E' dt



^"477 \dy dx £> dt)

1 /cZy8 da ]^dR\
E^ dt)



(112),



4Tr\dx dy E"

where p, q, r are the electric currents in the directions of x, y, and z; a, ^, y
are the components of magnetic intensity; and P, Q, R are the electromotive
forces. Now if e be the quantity of fi:ee electricity in unit of volume, then the
equation of continuity will be

I4M+S- (^^«)-

♦ See Rankine "On Elasticity," Camb. and Dub. Math. Joum. 1851.
t Phil Mag. March, 1861 [p. 462 of this vol.].



ON PHYSICAL LINES OF FORCE. 4^7

Differentiating (112) with respect to x, y, and z respectively, and substituting,

we find

de_J^d (JP clQ dR\

dt~ inE'dtydx dy dz j ^ "

1 (dP . dQ . dR\ .....

the constant being omitted, because e = when there are no electromotive forces.

Prop. XV. — To find the force acting between two electrified bodies.

The energy in the medium arising from the electric displacements is

U=-t^{Pf+Qg + Rh)hV (116),

where P, Q, R are the forces, and/, g, h the displacements. Now when there
is no motion of the bodies or alteration of forces, it appears from equations (77)*

that

j^ d^ ^ d^ j._ d^ , .

and we know by (105) that

P= ^AirElf, Q= -iirE'g, R=-i7rPPh (119);

dy\



^. 1 ^ fd^
whence ^=8^^ U



-^4-g)^^ (-).



Integi-ating by parts throughout all space, and remembering that ^ vanishes at
an infinite distance,

^-sk'H^.-w-^y (-)^

or by (115), U=it{^e)hV (122).

Now let there be two electrified bodies, and let e, be the distribution of
electricity in the first, and ^i the electric tension due to it, and let

1 fd^ d^ d^,\ , .

Let Cj be the distribution of electricity in the second body, and ^, the
tension due to it; then the whole tension at any point will be "^i + "*!'„ and
the expansion for U will become

ir=it(%e,-\-%e, + %e, + %e,)BV (124).

♦ PhU. Mag. May, 1861 [p. 482 of this vol.].

VOL. I. 63



498 ON PHYSICAL LINES OF FORCE.

Let the body whose electricity is e^ be moved in any way, the electricity
moving along with the body, then since the distribution of tension ^j moves
with the body, the value of %e^ remains the same.

%ei also remains the same; and Green has shewn (Essay on Electricity,
p. 10) that %e^ = %ei, so that the work done by moving the body against
electric forces

W=BU=S%(%e,)BV (125).

And if ei is confined to a small body,

W=e,B%,

or Fdr = e,'^dr (126),

where F is the resistance and dr the motion.

If the body e^ be small, then if ?' is the distance from e^, equation (123)
gives

r

whence F=-JS^%' (127);

or the force is a repulsion varying inversely as the square of the distance.

Now let 7)i and 172 be the same quantities of electricity measured stati-
cally, then we know by definition of electrical quantity

F=-'^ (128);

and this will be satisfied provided

ri, = Fe, and r), = Ee, (129);

so that the quantity F previously determined in Prop. XIII. is the number by
which the electrodynamic measure of any quantity of electricity must be
multipUed to obtain its electrostatic measure.

That electric current which, circulating round a ring whose area is unity,
produces the same efiect on a distant magnet as a magnet would produce



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