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can be found, identical with a mixture of the other two.

4th. These results may be stated in the form of colour-equations, giving
the numerical value of the amount of each colour entering into any mixture.
By means of the Colour Top'", such equations can be obtained for coloured
papers, and they may be obtained with a degree of accuracy shewing that the
colour-judgment of the eye may be rendered very perfect.

The speaker had tested in this way more than 100 different pigments and
mixtures, and had found the results agree with the theory of three primaries

* Described in the Trans, of the Royal Society of Edinburgh, Vol. xxi., and in the Phil. Mag.



ON THE THEORY OF THREE PRIMARY COLOURS. 449

in every case. He had also examined all the colours of the spectrum with
the same result.

The experiments with pigments do not indicate what colours are to be
considered as primary ; but experiments on the prismatic spectrum shew that
all the colours of the spectrum, and therefore all the colours in nature, are
equivalent to mixtures of three colours of the spectrum itself, namely, red,
green (near the line E), and blue (near the line G). Yellow was found to be
a mixture of red and green.

The speaker, assuming red, green, and blue as primary colours, then exhi-
bited them on a screen by means of three magic lanterns, before which were
placed glass troughs containing respectively sulphocyanide of iron, chloride of
copper, and ammoniated copper.

A triangle was thus illuminated, so that the pure colours appeared at its
angles, while the rest of the triangle contained the various mixtures of the
colours as in Young's triangle of colour.

The graduated intensity of the primary colours in different parts of the
spectrum was exhibited by three coloured images, which, when superposed on
the screen, gave an artificial representation of the spectrum.

Three photographs of a coloured ribbon taken through the three coloured
solutions respectively, were introduced into the camera, giving images represent-
ing the red, the green, and the blue parts separately, as they would be seen
by each of Young's three sets of nerves separately. When these were super-
posed, a coloured image was seen, which, if the red and green images had
been as fully photographed as the blue, would have been a truly-coloured image
of the ribbon. By finding photographic materials more sensitive to the less
refrangible rays, the representation of the colours of objects might be greatly
improved.

The speaker then proceeded to exhibit mixtures of the colours of the pure
spectrum. Light from the electric lamp was passed through a narrow slit, a
lens and a prism, so as to throw a pure spectrum on a screen containing three
moveable slits, through which three distinct portions of the spectrum were
suffered to pass. These portions were concentrated by a lens on a screen at
a distance, forming a large, uniformly coloured image of the prism.

When the whole spectrum was allowed to pass, this image was white, as
in Newton's experiment of combining the rays of the spectrum. When portions
of the spectrum were allowed to paas through the moveable slits, the image was

VOL. L 57



450 ON THE THEORY OF THREE PRIMARY COLOURS.

uniformly illuminated with a mixture of the corresponding colours. In order
to see these colours separately, another lens was placed between the moveable
slits and the screen. A magnified image of the sHts was thus thrown on the
screen, each sHt shewing, by its colour and its breadth, the quality and quantity
of the colour which it suffered to pass. Several colours were thus exhibited,
first separately, and then in combination. Red and blue, for instance, produced
purple ; red and green produced yellow ; blue and yellow produced a pale pink ;
red, blue, and green produced white; and red and a bluish green near the
line F produced a colour which appears very different to different eyes.

The speaker concluded by stating the peculiarities of colour-blind vision,
and by shewing that the investigation into the theory of colour is truly a
physiological inquiry, and that it requires the observations and testimony of
persons of every kind in order to discover and explain the various peculiarities
of vision.



[From the Philosophical Magazine, Vol. xxi.]



XXIII. On Physical Lines of Force.

PART I.
The Theory of Molecular Vortices applied to Magnetic Phenomena.

In all phenomena involving attractions or repulsions, or any forces depend-
ing on the relative position of bodies, we have to determine the magnitude and
direction of the force which would act on a given body, if placed in a given
position.

In the case of a body acted on by the gravitation of a sphere, this force
is inversely as the square of the distance, and in a straight line to the centre
of the sphere. In the case of two attracting spheres, or of a body not spherical,
the magnitude and direction of the force vary according to more complicated
laws. In electric and magnetic phenomena, the magnitude and direction of the
resultant force at any point is the main subject of investigation. Suppose that
the direction of the force at any point is known, then, if we draw a line so
that in every part of its course it coincides in direction with the force at that
point, this hne may be called a line of force, since it indicates the direction
of the force in every part of its course.

By drawing a sufficient number of lines of force, we may indicate the
direction of the force in every part of the space in which it acts.

Thus if we strew iron filings on paper near a magnet, each filing will be
magnetized by induction, and the consecutive filings will unite by their opposite
poles, so as to form fibres, and these fibres will indicate the direction of the lines
of force. The beautiful illustration of the presence of magnetic force afforded
by this experiment, naturally tends to make us think of the lines of force as
something real, and as indicating something more than the mere resultant of
two forces, whose seat of action is at a distance, and which do not exist there

57—2



452 ON PHYSICAL LINES OF FORCE.

at all until a magnet is placed in that part of the field. We are dissatisfied
with the explanation founded on the hypothesis of attractive and repellent
forces directed towards the magnetic poles, even though we may have satisfied
ourselves that the phenomenon is in strict accordance with that hypothesis, and
we cannot help thinking that in every place where we find these lines of force,
some physical state or action must exist in sufficient energy to produce the
actual phenomena.

My object in this paper is to clear the way for speculation in this direction,
by investigating the mechanical results of certain states of tension and motion
in a medium, and comparing these with the observed phenomena of magnetism
and electricity. By pointing out the mechanical consequences of such hypotheses,
I hope to be of some use to those who consider the phenomena as due to the
action of a medium, but are in doubt as to the relation of this hypothesis to
the experimental laws already established, which have generally been expressed
in the language of other hypotheses.

I have in a former paper* endeavoured to lay before the mind of the
geometer a clear conception of the relation of the lines of force to the space
in which they are traced. By making use of the conception of currents in a
fluid, I shewed how to draw lines of force, which should indicate by their
number the amount of force, so that each line may be called a unit-line of
force (see Faraday's Reswear dies, 3122); and I have investigated the path of
the lines where they pass from one medium to another.

In the same paper I have found the geometrical significance of the "Elec-
trotonic State," and have shewn how to deduce the mathematical relations
between the electrotonic state, magnetism, electric currents, and the electromotive
force, using mechanical illustrations to assist the imagination, but not to account
for the phenomena.

I propose now to examine magnetic phenomena from a mechanical point of
view, and to determine what tensions in, or motions of, a medium are capable
of producing the mechanical phenomena observed. If, by the same hypothesis,
we can connect the phenomena of magnetic attraction with electromagnetic phe-
nomena and with those of induced currents, we shall have found a theory
which, if not true, can only be proved to be erroneous by experiments which
will greatly enlarge our knowledge of this part of physics.

♦ See a paper " On Faraday's Lines of Force," Cambridge Philosophical Transactions, Vol. i. Part i.
Page 155 of this volume.



ON PHYSICAL LINES OF FORCE. 453

The mechanical conditions of a medium under magnetic influence have been
variously conceived of, as currents, undulations, or states of displacement or
strain, or of pressure or stress.

Currents, issuing from the north pole and entering the south pole of a
magnet, or circulating round an electric current, have the advantage of repre-
senting correctly the geometrical arrangement of the lines of force, if we could
account on mechanical principles for the phenomena of attraction, or for the
currents themselves, or explain their continued existence

Undulations issuing from a centre would, according to the calculations of
Professor Challis, produce an effect similar to attraction in the direction of the
centre ; but admitting this to be true, we know that two series of undulations
traversing the same space do not combine into one resultant as two attractions
do, but produce an effect depending on relations of phase as well as intensity,
and if allowed to proceed, they diverge from each other without any mutual
action. In fact the mathematical laws of attractions are not analogous in any
respect to those of undulations, while they have remarkable analogies with those
of currents, of the conduction of heat and electricity, and of elastic bodies.

In the Cambridge and Dublin Mathematical Journal for January 1847,
Professor William Thomson has given a "Mechanical Representation of Electric,
Magnetic, and Galvanic Forces," by means of the displacements of the particles of
an elastic solid in a state of strain. In this representation we must make the
angular displacement at every point of the solid proportional to the magnetic
force at the con-esponding point of the magnetic field, the direction of the axis
of rotation of the displacement corresponding to the direction of the magnetic
force. The absolute displacement of any particle will then correspond in magni-
tude and direction to that which I have identified with the electrotonic state ;
and the relative displacement of any particle, considered with reference to the
particle in its immediate neighbourhood, will correspond in magnitude and direc-
tion to the quantity of electric current passing through the corresponding point
of the magneto-electric field. The author of this method of representation does
not attempt to explain the origin of the observed forces by the effects due to
these strains in the elastic solid, but makes use of the mathematical analogies
of the two problems to assist the imagination in the study of both.

We come now to consider the magnetic influence as existing in the form of
some kind of pressure or tension, or, more generally, of stress in the medium.

Stress is action and reaction between the consecutive parts of a body, and



454 ON PHYSICAL LINES OF FORCE.

consists in general of pressures or tensions different in different directions at
the same point of the medium.

The necessary relations among these forces have been investigated by mathe-
maticians ; and it has been shewn that the most general type of a stress
consists of a eombmation of three principal pressures or tensions, in directions
at right angles to each other.

When two of the principal pressures are equal, the third becomes an axis
of symmetry, either of greatest or least pressure, the pressures at right angles
to this axis being all equal.

When the three principal pressures are equal, the pressure is equal in every
direction, and there results a stress having no determinate axis of direction, of
which we have an example in simple hydrostatic pressure.

The general type of a stress is not suitable as a representation of a mag^
netic force, because a line of magnetic force has direction and intensity, but
has no third quahty indicating any difference between the sides of the line,
which would be analogous to that observed in the case of polarized light*.

We must therefore represent the magnetic force at a point by a stress
having a single axis of greatest or least pressure, and all the pressures at right
angles to this axis equal. It may be objected that it is inconsistent to represent
a line of force, which is essentially dipolar, by an axis of stress, which is
necessarily isotropic; but we know that every phenomenon of action and reaction
is isotropic in its results, because the effects of the force on the bodies between
which it acts are equal and opposite, while the nature and origin of the force
may be dipolar, as in the attraction between a north and a south pole.

Let us next consider the mechanical effect of a state of stress symmetrical
about an axis. We may resolve it, in all cases, into a simple hydrostatic
pressure, combined with a simple pressure or tension along the axis. When the
axis is that of greatest pressure, the force along the axis will be a pressure.
When the axis is that of least pressure, the force along the axis will be a
tension.

K we observe the lines of force between two magnets, as indicated by iron
filings, we shall see that whenever the Hnes of force pass firom one pole to
another, there is attraction between those poles; and where the lines of force
from the poles avoid each other and are dispersed into space, the poles repel

* See Faraday's Researches, 3262.



ON PHYSICAL LINES OF FORCE. 455

each other, so that in both cases they are drawn in the direction of the
resultant of the lines of force.

It appears therefore that the stress in the axis of a line of magnetic force
is a tension, like that of a rope.

If we calculate the lines of force in the neighbourhood of two gravitating
bodies, we shall find them the same in direction as those near two magnetic
poles of the same name ; but we know that the mechanical effect is that of
attraction instead of repulsion. The lines of force in this case do not run
between the bodies, but avoid each other, and are dispersed over space. In
order to produce the effect of attraction, the stress along the lines of gravi-
tating force must be a pressure.

Let us now suppose that the phenomena of magnetism depend on the
existence of a tension in the direction of the lines of force, combined with a
hydrostatic pressure; or in other words, a pressure greater in the equatorial
than in the axial direction : the next question is, what mechanical explanation
can we give of this inequality of pressures in a fluid or mobUe medium ? The
explanation which most readily occurs to the mind is that the excess of pres-
sure in the equatorial direction arises from the centrifugal force of vortices or
eddies in the medium having their axes in directions parallel to the lines of force.

This explanation of the cause of the inequality of pressures at once suggests
the means of representing the dipolar character of the line of force. Every
vortex is essentially dipolar, the two extremities of its axis being distinguished
by the direction of its revolution as observed from those points.

We also know that when electricity circulates in a conductor, it produces
lines of magnetic force passing through the circuit, the direction of the lines
depending on the direction of the circulation. Let us suppose that the direction
of revolution of our vortices is that in which vitreous electricity must revolve
in order to produce lines of force whose direction within the circuit is the
same as that of the given lines of force.

We shall suppose at present that all the vortices in any one part of the
field are revolving in the same direction about axes nearly parallel, but
that in passing from one part of the field to another, the direction of the
axes, the velocity of rotation, and the density of the substance of the vortices
are subject to change. We shall investigate the resultant mechanical effect upon
an element of the medium, and from the mathematical expression of this
resultant we shall deduce the physical character of its different component parts.



456 ON PHYSICAL LINES OF FORCE.

Prop. I. — If in two fluid systems geometrically similar the velocities and
densities at corresponding points are proportional, then the differences of pres-
sure at corresponding points due to the motion will vary in the duplicate ratio
of the velocities and the simple ratio of the densities.

Let I be the ratio of the linear dimensions, m that of the velocities,
n that of the densities, and p that of the pressures due to the motion. Then
the ratio of the inasses of corresponding portions will be Vn, and the ratio of
the velocities acquired in traversing similar parts of the systems will be m ;
so that l^mn is the ratio of the momenta acquired by similar portions in
traversing similar parts of their paths.

The ratio of the surfaces is P, that of the forces acting on them is I'^p,

and that of the times during which they act is — ; so that the ratio of the
impulse of the forces is — , and we have now

m
or m^n =jp ;

that is, the ratio of the pressures due to the motion (p) is compounded of
the ratio of the densities (n) and the duplicate ratio of the velocities {ni"), and
does not depend on the linear dimensions of the moving systems.

In a circular vortex, revolving with uniform angular velocity, if the
pressure at the axis is p^, that at the circumference will be i>i=jPo + ip^j where
p is the density and v the velocity at the circumference. The mean pressure
parallel to the axis will be

If a number of such vortices were placed together side by side with their
axes parallel, they would form a medium in which there would be a pressure
Pz parallel to the axes, and a pressure p^ in any perpendicular direction. If the
vortices are circular, and have uniform angular velocity and density throughout,
then

Pi-P2 = lp'^'
If the vortices are not circular, and if the angular velocity and the density
are not uniform, but vary according to the same law for all the vortices,

Pi-p.^Cpif,



ON PHYSICAL LINES OF FORCE. 457

where p is the mean density, and C is a numerical quantity depending on the
distribution of angular velocity and density in the vortex. In future we shall

write -7^ instead of Co, so that

477- '^

^'"^'^4^''''' (^)'

where /n is a quantity bearing a constant ratio to the density, and v is the
linear velocity at the circumference of each vortex.

A medium of this kind, filled with molecular vortices having their axes
parallel, differs from an ordinary fluid in having different pressures in different
directions. If not prevented by properly arranged pressures, it would tend to
expand laterally. In so doing, it would allow the diameter of each vortex to
expand and its velocity to diminish in the same proportion. In order that a
medium having these inequalities of pressure in different directions should be in
equihbrium, certain conditions must be fulfilled, which we must investigate.

Prop. II. — If the direction-cosines of the axes of the vortices with respect
to the axes of x, y, and z be /, m, and n, to find the normal and tangential
stresses on the co-ordinate planes.

The actual stress may be resolved into a simple hydrostatic pressure p^ acting
in all directions, and a simple tension Pi—p^, or -7- fiif, acting along the axis
of stress.

Hence if p^x, pyy, and p^ be the normal stresses parallel to the three axes,
considered positive when they tend to increase those axes ; and if p^^, p^, and
Pj^ be the tangential stresses in the three co-ordinate planes, considered positive
when they tend to increase simultaneously the symbols subscribed, then by
the resolution of stresses*,

Pxx = j^l^vn'-p„
1 . ,

* Rankine's Applied Mechanics, Art. 106.
VOL. I. 58



458



ON PHYSICAL LINES OF FORCE.



If we write
then



a = vl, ^ = vm, and y = vn,






Air
1






(2).



Prop. III. — To find the resultant force on an element of the medium,
arising from the variation of internal stress.

"We have in general, for the force in the direction of x per unit of volume
by the law of equilibrium of stresses*,

V d d d ,„v

^'TxP-'+TyP-' + dzP' (^)-

In this case the expression may be written

Remembering that a ^ + /8 ^ + y ^ = i ^ (a" + jff + y"), this becomes

. I ld& da.\ _ 1 Ida. dy\ dp, , ,

-l'^i^[di-Ty)+l'->'Tn[di-di)-dS - -^^'-

The expressions for the forces parallel to the axes of y and z may be written
down from analogy.

* Baiikine's Applied MecJianics, Art. 116.



ON PHYSICAL LINES OF FORCE. 459

We have now to interpret the meaning of each term of this expression.
We suppose a, /3, y to be the components of the force which would act
upon that end of a unit magnetic bar which points to the north.

/x represents the magnetic inductive capacity of the medium at any point
referred to air as a standard, /la, /i,/3, /xy represent the quantity of magnetic
induction through unit of area perpendicular to the three axes of x, y z
respectively.

The total amount of magnetic induction through a closed surface surrounding
the pole of a magnet, depends entirely on the strength of that pole ; so that
if dxdydz be an element, then

(-T-/xa + -i-/>t/3 + -T- /lyj dxdydz = i'rrm dxdydz (6),

which represents the total amount of magnetic induction outwards through the
surface of the element dxdydz, represents the amount of "imaginary magnetic
matter" within the element, of the kind which points north.

The first term of the value of X, therefore,

1 /d d n d \ /_.

''ii[dx''^ + d^l'^ + dz''V (^)'

may be written

am (8),

where a is the intensity of the magnetic force, and m is the amount of mag-
netic matter poLnting north in unit of volume.

The physical interpretation of this term is, that the force urging a north pole
in the positive direction of a; is the product of the intensity of the magnetic
force resolved in that direction, and the strength of the north pole of the magnet.

Let the parallel lines from left to right in fig. 1 represent a field of mag-
netic force such as that of the earth, sn being the direction from south to north.
The vortices, according to our hypothesis, will be in the direction shewn by the
arrows in fig. 3, that is, in a plane perpendicular to the lines of force, and
revolving in the direction of the hands of a watch when observed from 5
looking towards n. The parts of the vortices above the plane of the paper
will be moving towards e, and the parts below that plane towards w.

58—2



460



ON PHYSICAL LINES OF FORCE.



Fig. 1.



1^


^


\ k/




(^c r


"^ ^TJ


"ll


t<: ^


1 ^^




/ X


N


' "






Fig. 2.

1




V


/


^


\n


'^




S B >@-^ Y


.^


/•


^\




/


\


^



We shall always mark by an arrow-head the direction in which we must
look in order to see the vortices rotating in the
direction of the hands of a watch. The arrow-head
will then indicate the northward direction in the
magnetic field, that is, the direction in which that
end of a magnet which points to the north would
set itself in the field.

Now let A be the end of a magnet which
points north. Since it repels the north ends of
other magnets, the Hues of force wiU be directed
from A outwards in all directions. On the north
side the line AD wiU be in the sarae direction with
the lines of the magnetic field, and the velocity of
the vortices will be increased. On the south side
the line AC will be in the opposite direction, and
the velocity of the vortices wUl be diminished, so
that the lines of force are more powerful on the
north side of A than on the south side.

We have seen that the mechanical efiect of the
vortices is to produce a tension along their axes,
so that the resultant effect on A will be to pull

it more powerfully towards D than towards C\ that is, A will tend to move
to the north.

Let B in fig. 2 represent a south pole. The lines of force belonging to B
will tend towards B, and we shall find that the lines of force are rendered
stronger towards E than towards F, so that the effect in this case is to urge B
towards the south.

It appears therefore that, on the hypothesis of molecular vortices, our first



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