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an apparatus similar to that described by M. Foucault§, by which a screen of
white paper is illuminated by the mixed light. The field of mixed colour is
much larger than in M. Helmholtz's former experiments, and the facility of
forming combinations is much increased. In this memoir the mathematical theory
of Newton's circle, and of the curve formed by the spectrum, with its possible
transformations, is completely stated, and the form of this curve is in some
degree indicated, as far as the determination of the colours which he on oppo-
site sides of white, and of those which He opposite the part of the curve which
is wanting. The colours between red and yellow-green are complementary to
colours between blue-green and violet, and those between yellow-green and blue-
green have no homogeneous complementaries, but must be neutrahzed by various
hues of purple, i.e., mixtures of red and violet. The names of the complementary
colours, with their wave-lengths in air, as deduced from Fraunhofer's measure-
ments, are given in the following table : —

• PoggendorflF's Annalen, BA xciv. (I am indebted for the perusal of this Memoir to Professor

+ lb. Bd. Lxxxvii. Annals of Philosophy, 1852, Part ii.

t Ih. Bd. Lxxxix. Ann. Phil., 1854, April.

§ lb. Bd. LXixvm. Moigno, Cosmos, 1853, Tom. ii,, p. 232.







Ratio of

Red ... .

Orange . . .
Gold veUow .
Yellow . . .
Yellow . . .
Green-yellow .


Green-blue .
Blue . . .
Blue . . .
Blue . . .
Violet . .



(The wave-lengtha are expressed in millionths of a Paris inch.)

(In order to reduce these wave-lengths to their actual lengths in the eye,
each must be divided by the index of refraction for that kind of light in the
medium in which the physical etfect of the vibrations is supposed to take place.)

Although these experiments are not in themselves sufficient to give the com-
plete theory of the curve of homogeneous colours, they determine the most
important element of that theory in a way which seems very accurate, and I
cannot doubt that when a philosopher who has so fully pointed out the im-
portance of general theories in physics turns his attention to the theory of
sensation, he will at least establish the principle that the laws of sensation can
be successfully investigated only after the corresponding physical laws have been
ascertained, and that the connection of these two kinds of laws can be appre-
hended only when the distinction between them is fully recognised.

Note IV.

Description of the Figures. Plate I.

No. 1. is the colour-diagi-am already referred to, representing, cm Newton's principle, the relations of
diflferent coloured papers to the three standard colours— vermilion, emerald-green, and ultra-
marine. The initials denoting the colours are explained in the list at page 276, and the
numbers belonging to them are their coefficients of intensity, the use of which has been
explained. The initials H.R., H.B., and H.G., represent the red, blue and green papers
of Mr Hay, and serve to connect this diagram -vith No. (2), which takes these colours for
its standards.
VOL. I. 20


No. 2. represents the relations of Mr Hay's red, blue, green, white, and yellow papers, as deter-
mined by a large number of experiments at Cambridge. — (See Note II.). The use of the
point D, in calculating the results of colour-blindness, is explained in the Paper.

Fig, 3. represents a disc of the larger size, with its slit.

Fig. 4. shows the mode of combining two discs of the smaller size.

Fi«^. 5. shows the combination of discs, as placed on the top, in the first experiment described

in the Paper.
Fig. 6. represents the method of spinning the top, when speed is required.
The last four figures are half the actual size.

Colour-tops of the kind used in these experiments, with paper discs of the colours whose relations
are represented in No. 1, are to be had of Mr J. M. Bryson, Optician, Edinburgh.








HB06* i


^GO-2 /












FIG. 6

[From the Transactions of the Cambridge Philosophical Society, VoL x. Part i.]

VIII. On Faraday's Lines of Force.
[Read Dec. 10, 1855, and Feb. 11, 1856.]

The present state of electrical science seems peculiarl^^ unfavourable to specu-
l ation. The laws of the distribution of electricity on the surface of conductors
have been analytically deduced from experiment; some parts of the mathematical
theory of magnetism are established, while in other parts the experimental data
are wanting ; the theory of the conduction of galvan ism and that of the mutual
attraction of conductors have been reduced to mathematical formulae, but have
not fallen into relation with the other parts of the science. No electrical theory
can now be put forth, unless it shews the connexion not only between electricity
at rest and current electricity, but between the attractions and inductive effects
of electricity in both states. Such a theory must accurately satisfy those laws,
the mathematical form of which is known, and must afford the means of calcu-
lating the effects in the limiting cases where the known formulae are inapplicable.
In order therefore to appreciate the requirements of the science, the student
must make himself familiar with a considerable body of most intricate mathe-
matics, the mer fi retention of which in the memory materially interferes with
further progress. The first process therefore in the effectual study of the science^,
must be one of simplification and reduction of the results of previous investiga-
tion to a form in which the mind can grasp them. The results of this simplifi-
cation may take the form of a purely mathematical formula or of a physical
hypothesis. In the first case we entirely lose sight of the phenomena to be
explained ; and though we may trace out the consequences of given laws, we
can never obtain more extended views of the connexions of the subject^ If,
on the other luiml, we adopt a physical hypothesis, we see the phenomena only
throucrh a medium, and are liable to that blindness to facts and rashness m



assumption wKich a partial explanation encourages. "We must therefore discover
some method of investigation which allows the mind at every step to lay hold
of a clear physical conception, without being committed to any theory founded
on the physical science from which that conception is borrowed, so that it is
neither drawn aside from the subject in pursuit of analytical subtleties, nor carried
beyond the truth by a favourite hypothesis.

In order to obtain physical ideas without adopting a physical theory we must
make ourselves familiar with the existence of physical analogies. By a physical
analogy I mean that partial similarity between the laws of one science and those
of another which makes each of them illustrate the other. Thus all the mathe-
matical sciences are founded on relations between physical laws and laws of
numbers, so that the aim of exact science is to reduce the problems of nature
to the determination of quantities by operations with numbers. Passing from
the most universal of all analogies to a very partial one, we find the same
resemblance in mathematical form between two different phenomena giving rise
to a physical theory of light.

The changes of direction which light undergoes in passing from one medium
to another, are identical with the deviations of the path of a particle in moving
through a narrow space in which intense forces act. This analogy, which extends
only to the direction, and not to the velocity of motion, was long believed to
he the true explanation of the refraction of Ught ; and we still find it useful
in the solution of certain problems, in which we employ it without danger, as
an artificial method. The other analogy, between light and the vibrations of an
elastic medium, extends much farther, but, though its importance and fruitfulness
cannot be over-estimated, we must recollect that it is founded only on a resem-
blance in form between the laws of light and those of vibrations. By stripping
it of its physical dress and reducing it to a theory of " transverse alternations,"
we might obtain a system of truth strictly founded on observation, but probably
deficient both in the vividness of its conceptions and the fertility of its method.
I have said thus much on the disputed questions of Optics, as a preparation
for the discussion of the almost universally admitted theory of attraction at a

We have all acquired the mathematical conception of these attractions. We
can reason about them and determine their appropriate forms or formulae. These
formulae have a distinct mathematical significance, and their results are found
to be in accordance with natural phenomena. There is no formula in applied

ON FARADAY'3 lines OF FORCE. 157

mathematics more consistent with nature than the formula of attractions, and no
theory better estabUshed in the minds of men than that of the action of bodies
on one another at a distance. The laws of the conduction of heat in uniform
media appear at first sight among the most different in their physical relations
from those relating to attractions. The quantities which enter into them are
teviperature, flow of heat, conductivity. The word force is foreign to the subject.
Yet we find that the mathematical laws of the uniform motion of heat in
homogeneous media are identical in form with those of attractions varying in-
versely as the square of the distance. We have only to substitute source of
heat for centre of attrax^tion, flow of heat for accelerating effect of attraction at
any point, and temperature for potential, and the solution of a problem in
attractions is transformed into that of a problem in heat.

This analogy between the formulae of heat and attraction was, I believe,
first pointed out by Professor William Thomson in the Camh. Math. Journal,
Vol. III.

Now the conduction of heat is supposed to proceed by an action between
contiguous parts of a medium, while the force of attraction is a relation be-
tween distant bodies, and yet, if we knew nothing more than is expressed in
the mathematical formulae, there would be nothing to distinguish between the
one set of phenomena and the other.

It is true, that if we introduce other considerations and observe additional
facts, the two subjects will assume very difierent aspects, but the mathematical
resemblance of some of their laws will remain, and may still be made useful
in exciting appropriate mathematical ideas.

It is by the use of analogies of this kind that I have attempted to bring
before the mind, in a convenient and manageable form, those mathematical ideas
which are necessary to the study of the phenomena of electricity. The methods
are generally those suggested by the processes of reasoning which are found in
the researches of Faraday"*', and which, though they have been interpreted
mathematically by Prof. Thomson and others, are very generally supposed to be
of an indefinite and unmathematical character, when compared with those em-
ployed by the professed mathematicians. By the method which I adopt, I hope
to render it evident that I am not attempting to estabhsh any physical theory
of a science in which I have hardly made a single experiment, and that the
limit of my design is to shew how, by a strict application of the ideas and

* See especially Series xxxviii. of the Experimental Researcltes, and Phil. Mag. 1852.

158 ON Faraday's lines of force.

methods of Faraday, the connexion of the very different orders of phenomena
which he has discovered may be clearly placed before the mathematical mind.
I shall therefore avoid as much as I can the introduction of anything which
does not serve as a direct illustration of Faraday's methods, or of the mathe-
matical deductions which may be made from them. In treating the simpler
parts of the subject I shall use Faraday's mathematical methods as well as
his ideas. When the complexity of the subject requires it, I shall use analytical
notation, still confining myself to the development of ideas originated by the
same philosopher.

I have in the first place to explain and illustrate the idea of "lines of

When a body is electrified in any manner, a small body charged with posi-
tive electricity, and placed in any given position, will experience a force urging
it in a certain direction. If the small body be now negatively electrified, it will
be urged by an equal force in a direction exactly opposite.

The same relations hold between a magnetic body and the north or south
poles of a small magnet. If the north pole is urged in one direction, the south
pole is urged in the opposite direction.

In this way we might find a line passing through any point of space, such
that it represents the direction of the force acting on a positively electrified
particle, or on an elementary north pole, and the reverse direction of the force
on a negatively electrified particle or an elementary south pole. Since at every
point of space such a direction may be found, if we commence at any point
and draw a line so that, as we go along it, its direction at any point shall
always coincide with that of the resultant force at that point, this curve wiU
indicate the direction of that force for every point through which it passes, and
might be called on that account a line of force. We might in the same way
draw other lines of force, till we had filled all space with curves indicating by
their direction that of the force at any assigned point.

We should thus obtain a geometrical model of the physical phenomena,
which would tell us the direction of the force, but we should stiU require some
method of indicating the intensity of the force at any point. If we consider
these curves not as mere lines, but as fine tubes of variable section carrying
an incompressible fluid, then, since the velocity of the fluid is inversely as the
section of the tube, we may make the velocity vary according to any given law,
by regulating the section of the tube, and in this way we might represent the



intensity of the force as well as its direction by the motion of the fluid in
these tubes. This method of representing the intensity of a force by the velocity
of an imaginary fluid in a tube is applicable to any conceivable system of forces,
but it is capable of great simplification in the case in which the forces are such
as can be explained by the hypothesis of attractions varying inversely as the
square of the distance, such as those observed in electrical and magnetic pheno-
mena. In the case of a perfectly arbitrary system of forces, there will generally
be interstices between the tubes ; but in the case of electric and magnetic forces
it is possible to arrange the tubes so as to leave no interstices. The tubes will
then be mere surfaces, directing the motion of a fluid filling up the whole space.
It has been usual to commence the investigation of the laws of these forces by
at once assuming that the phenomena are due to attractive or repulsive forces
acting between certain points. We may however obtain a different view of the
subject, and one more suited to our more difficult inquiries, by adopting for the
definition of the forces of which we treat, that they may be represented in
magnitude and direction by the uniform motion of an incompressible fluid.

I propose, then, first to describe a method by which the motion of such a
fluid can be clearly conceived; secondly to trace the consequences of assuming
certain conditions of motion, and to point out the application of the method to
some of the less complicated phenomena of electricity, magnetism, and galvanism ;
and lastly to shew how by an extension of these methods, and the introduction
of another idea due to Faraday, the laws of the attractions and inductive actions
of magnets and currents may be clearly conceived, without making any assump-
tions as to the physical nature of electricity, or adding anything to that which
has been already proved by experiment.

By referring everything to the purely geometrical idea of the motion of an
imaginary fluid, I hope to attain generahty and precision, and to avoid the
dangers arising from a premature theory professing to explain the cause of the
phenomena. If the results of mere speculation which I have collected are found
to be of any use to experimental philosophers, in arranging and interpreting
their results, they will have served their purpose, and a mature theory, in which
physical facts will be physically explained, will be formed by those who by
interrogating Nature herself can obtain the only true solution of the questions
which the mathematical theory suggests.


I. Theoi-y of the Motion of an incompressible Fluid.

(1) The substance here treated of must not be assumed to possess any of
the properties of ordinary fluids except those of freedom of motion and resistance
to compression. It is not even a hypothetical fluid which is introduced to
explain actual phenomena. It is merely a collection of imaginary properties
which may be employed for establishing certain theorems in pure mathematics in
a way more intelligible to many minds and more applicable to physical problems
than that in which algebraic symbols alone are used. The use of the word
"Fluid" will not lead us into error, if we remember that it denotes a purely
imaginary substance with the following property :

The poHion of fluid which at any iTistant occupied a given volume, will at
any succeeding instant occupy an equal volume.

This law expresses the incompressibility of the fluid, and furnishes us with
a convenient measure of its quantity, namely its volume. The unit of quantity
of the fluid will therefore be the unit of volume.

(2) The direction of motion of the fluid will in general be dlflerent at
different points of the space which it occupies, but since the direction is deter-
minate for every such point, we may conceive a line to begin at any point and
to be continued so that every element of the line indicates by its direction the
direction of motion at that point of space. Lines drawn in such a manner that
their direction always indicates the direction of fluid motion are called lines of
fluid motion.

If the motion of the fluid be what is called steady motion, that is, if the
direction and velocity of the motion at any fixed point be independent of the
time, these curves will represent the paths of individual particles of the fluid,
but if the motion be variable this will not generally be the case. The cases
of motion which will come under our notice will be those of steady motion.

(3) If upon any surface which cuts the lines of fluid motion we draw a
closed curve, and if from every point of this curve we draw a line of motion,
these lines of motion will generate a tubular surface which we may call a tube
of fluid motion. Since this surface is generated by lines in the direction of fluid

ON Faraday's lines of force. 161

motion no part of the fluid can flow across it, so that this imaginary surface
is as impermeable to the fluid as a real tube.

(4) The quantity of fluid which in unit of time crosses any fixed section
of the tube is the same at whatever part of the tube the section be taken.
For the fluid is incompressible, and no part runs through the sides of the tube,
therefore the quantity which escapes from the second section is equal to that
which enters through the first.

If the tube be such that unit of volume passes through any section in
unit of time it is called a unit tube of fluid motion.

(5) In what follows, various units will be referred to, and a finite number
of lines or surfaces will be drawn, representing in terms of those units the
motion of the fluid. Now in order to define the motion in every part of the
fluid, an infinite number of lines would have to be drawn at indefinitely small
intervals ; but since the description of such a system of lines would involve
continual reference to the theory of limits, it has been thought better to suppose
the lines drawn at intervals depending on the assumed unit, and afterwards to
assume the unit as small as we please by taking a small submultiple of the
standard unit.

(6) To define the motion of the whole fluid by means of a system of unit

Take any fixed surface which cuts all the lines of fluid motion, and draw
upon it any system of curves not intersecting one another. On the same surface
draw a second system of curves intersecting the first system, and so arranged
that the quantity of fluid which crosses the surface within each of the quadri-
laterals formed by the intersection of the two systems of curves shall be unity
in unit of time. From every point in a curve of the first system let a line
of fluid motion be drawn. These lines will form a surface through which no
fluid passes. Similar impermeable surfaces may be drawn for all the curves of
the first system. The curves of the second system will give rise to a second
system of impermeable surfaces, which, by their intersection with the first system,
will form quadrilateral tubes, which will be tubes of fluid motion. Since each
quadrilateral of the cutting surface transmits unity of fluid in unity of time,
every tube in the system will transmit unity of fluid through any of its sections
in unit of time. The motion of the fluid at every part of the space it occupies

VOL, I. 21


is determined by this system of unit tubes ; for the direction of motion is that
of the tube through the point in question, and the velocity is the reciprocal
of the area of the section of the unit tube at. that point.

(7) We have now obtained a geometrical construction which completely
defines the motion of the fluid by dividing the space it occupies into a system
of unit tubes. We have next to shew how by means of these tubes we may
ascertain various points relating to the motion of the fluid.

A unit tube may either return into itself, or may begin and end at differ-
ent points, and these may be either in the boundary of the space in which we
investigate the motion, or within that space. In the first case there is a con-
tinual circulation of fluid in the tube, in the second the fluid enters at one end
and flows out at the other. If the extremities of the tube are in the bound-
ing surface, the fluid may be supposed to be continually supplied from without
from an unknown source, and to flow out at the other into an unknown reser-
voir ; but if the origin of the tube or its termination be within the space under
consideration, then we must conceive the fluid to be supplied by a source within
that space, capable of creating and emitting unity of fluid in unity of time, and
to be afterwards swallowed up by a sink capable of receiving and destroying
the same amount continually.

There is nothing self-contradictory in the conception of these sources where
the fluid is created, and sinks where it is annihilated. The properties of the
fluid are at our disposal, we have made it incompressible, and now we suppose
it produced from nothing at certain points and reduced to nothing at others.
The places of production will be called sources, and their numerical value will be
the number of units of fluid which they produce in unit of time. The places
of reduction will, for want of a better name, be called sinks, and will be esti-
mated by the number of units of fluid absorbed in unit of time. Both places
win sometimes be called sources, a source being understood to be a sink when
its sign is negative.

(8) It is evident that the amount of fluid which passes any fixed surface
is measured by the number of unit tubes which cut it, and the direction in
which the fluid passes is determined by that of its motion in the tubes. If
the surface be a closed one, then any tube whose terminations lie on the same
side of the surface must cross the surface as many times in the one direction
as in the other, and therefore must cany as much fluid out of the surface as

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