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Ail' by our calculation 40000

It appears, therefore, that the resistance of a stratum of air to the con-
duction of heat is about 10,000,000 times greater than that of a stratum of



ILLUSTRATIONS OF THE DYNAMICAL THEORY OF GASES. 405

copper of equal thickness. It would be almost impossible to establish the value
of the conductivity of a gas by direct experiment, as the heat radiated from the
sides of the vessel would be far greater than the heat conducted through the
air, even if currents could be entirely prevented*.



PART III.

ON THE COLLISION OF PERFECTLY ELASTIC BODIES OF ANY FORM.

When two perfectly smooth spheres strike each other, the force which acts
between them always passes through their centres of gravity ; and therefore their
motions of rotation, if they have any, are not affected by the collision, and
do not enter into our calculations. But, when the bodies are not spherical,
the force of compact will not, in general, be in the line joining their centres
of gravity ; and therefore the force of impact will depend both on the motion
of the centres and the motions of rotation before impact, and it will affect
both these motions after impact. .

In this way the velocities of the centres and the velocities of rotation
will act and react on each other, so that finally there will be some relation
established between them ; and since the rotations of the particles about their
three axes are quantities related to each other in the same way as the three
velocities of their centres, the reasoning of Prop. IV. will apply to rotation as
well as velocity, and both will be distributed according to the law

dN ^r 1 -
-T- = i V — j^ e *' .
ax a. 'Ju

* [Clausius, in the memoir cited in the last foot-note, has pointed out two oversights in this
calculation. In the first place the numbers have not been proi^erly reduced to English measure,
and have still to be multiplied by 4356, the ratio of the English pound to the kilogramme. The
numbers have, further, been calculated with one hour as the unit of time, whereas Maxwell h>\s
used them as if a second had been the unit. Taking account of these circumstarces and using his
own expression for the conduction which differs from (59) only in haNnng ^V in place of ^ on the
right-hand side, Clausius finds that the resistance of a stratum of air to the conduction of heat is
1400 times greater than that of a stratum of lead of the same thickness, or about 7000 times greater
than that of copper. Ed.]



406 ILLUSTRATIONS OF THE DYNAMICAL THEORY OF GASES.

Also, by Prop. V., if a; be tbe average velocity of one set of particles, and y
that of another, then the average value of the sum or difference of the velocities is

from which it is easy to see that, if in each individual case

w = ax + fey + cz,

where x, y, z are independent quantities distributed according to the law above
stated, then the average values of these quantities will be connected by the
equation

Prop. XXII. Two perfectly elastic bodies of any form strike each other:
given their motions before impact, and the line of i^npact, to find their motions
after impact.

Let M, and M, be the centres of gravity of the two bodies. M,X„ M,Y„
and i¥jZ, the principal axes of the first; and MJC^,
M,Y, and M^, those of the second. Let / be the
point of impact, and EJE, the line of impact.

Let the co-ordinates of / with respect to if, be
x^,z„ and with respect to M^ let them be x.^.jt,.

Let the direction-cosines of the line of impact
RJR, be l,m,n, with respect to M„ and l,7n,n, with
respect to M^.

^ Let M, and M, be the masses, and A.B^ and A,BA the moments of
inertia of the bodies about their principal axes.

Let the velocities of the centres of gravity, resolved in the direction of
the principal axes of each body, be

Z7„ F„ W„ and U,, V„ Tr„ before impact,
^^^ ^» y» W\, and ir„ F„ W'„ after impact.

Let the angular velocities round the same axes be

Pi, q^ r„ and p„ q„ r„ before impact,
^^^ P\> ?'i. f^'i, and p\, q\, r^ after impact.




ILLUSTRATIONS OF THE DYNAMICAL THEORY OF GASES. 407

Let R be the impulsive force between the bodies, measured by the momentum
it produces in each.

Then, for the velocities of the centres of gravity, we have the following
equations :

^'■= ^'+f'' ^'•= ^'-K (^2),

with two other pairs of equations in V and W.
The equations for the angular velocities are

p\ =Pi + -J (y^n, - z,m,), p, =p, - -J (y,n, - z,m,) (63),

with two other pairs of equations for q and r.

The condition of perfect elasticity is that the whole vis viva shall be the
same after impact as before, which gives the equation

M, ( U\ - U\) + M, ( U'\ - U\) + A, {p\ -p\) + A, {p\ -p\) + &c. = 0. . . . (64).

The terms relating to the axis of x are here given ; those relating to y and
z may be easily written down.

Substituting the values of these terms, as given by equations (62) and (63),
and dividing by R, we find

h{U\+ U,)-k{U\+ U,) + (y,n,-z,m,)(p\+p,)-{y,n,-z,m,) (p\+p,) + &c. = 0...{e5).

Now if v^ be the velocity of the striking-point of the first body before
impact, resolved along the line of impact,

v^ = lJJ^-\- (y^Tii — z^mi) pi + &c. ;

and if we put v^ for the velocity of the other striking-point resolved along the
same line, and v\ and v\ the same quantities after impact, we may write,
equation (65),

v^-\-v\ — v^ — v\ = (66),

or v^-Vj = v\-v\ (67),

which shows that the velocity of separation of the striking-points resolved in
the line of impact is equal to that of approach.



408 ILLUSTRATIONS OF THE DYNAMICAL THEORY OF GASES.

Substituting the values of the accented quantities in equation (65) by means
of equations (63) and (64), and transposing terms in J?, we find
2 {UJ, - UJ, +Pi {y,n, - z,m,) -p, {y,n, - zjn,)} 4- &c.

the other terms being related to y and z as these are to x. From this equation
we may find the value of E ; and by substituting this in equations (63), (64),
we may obtain the values of all the velocities after impact.

"We may, for example, find the value of U\ from the equation

ir (^' , 4' , {y.n,-z,m,Y . {y.n.-z.'m^Y ] M, ]

^^\M^M^ A, + A, ^^7T

-^a M^M^ — A — ^ — A — "^^'TT

+ 2 U,l, - 2p, {y,n, - z,m,) + 2p, (y^i, - z,m,) - &c.



(69).



Prop. XXIII. To find the relations between the average velocities of trans-
lation and rotation after many collisions among many bodies.

Taking equation (69), which applies to an individual collision, we see that
U\ is expressed as a linear function of Z7„ U„ p„ p„ &c., all of which are
quantities of which the values are distributed among the different particles
according to the law of Prop. IV. It follows from Prop. V., that if we square
every term of the equation, we shall have a new equation between the average
values of the different quantities. It is plain that, as soon as the required
relations have been estabUshed, they will remain the same after collision, so that
we may put Z7;"= U,' in the equation of averages. The equation between the
average values may then be written

Now since there are collisions in every possible way, so that the values of
I, m, n, &c. and x, y, z, &c. are infinitely varied, this equation cannot subsist
unless

The final state, therefore, of any number of systems of moving particles of
any form is that in which the average vis viva of translation along each of the



ILLUSTRATIONS OF THE DYNAMICAL THEORY OF GASES. 409

three axes ia the same in all the systems, and equal to the average vis viva
of rotation about each of the three principal axes of each particle.

Adding the tires vivcn with respect to the other axes, we find that the
whole via viva of translation is equal to that of rotation in each system of
particles, and is also the same for different systems, as was proved in Prop. VI.

This result (which is true, however nearly the bodies approach the spherical
form, provided the motion of rotation is at all afiected by the collisions) seems
decisive against the unqualified acceptation of the hypothesis that gases are such
systems of hard elastic particles. For the ascertained fact that y, the ratio of
the specific heat at constant pressure to that at constant volume, is equal to
1-408, requires that the ratio of the whole vis viva to the vis viva of translation
should be

^ = 3(^) = '-«^^^
whereas, according to our hypothesis, )S = 2 .

We have now followed the mathematical theory of the collisions of hard
elastic particles through various cases, in which there seems to be an analogy
with the phenomena of gases. We have deduced, as others have done already,
the relations of pressure, temperature, and density of a single gas. We have
also proved that when two different gases act freely on each other (that is, when
at the same temperature), the mass of the single particles of each is inversely
proportional to the square of the molecular velocity ; and therefore, at equal
temperature and pressure, the number of particles in unit of volume is the same.

We then offered an explanation of the internal friction of gases, and deduced
from experiments a value of the mean length of path of a particle between
successive collisions.

We have applied the theory to the law of diffusion of gases, and, from an
experiment on olefiant gas, we have deduced a value of the length of path not
very different from that deduced from experiments on friction.

Using this value of the length of path between collisions, we found that the
resistance of air to the conduction of heat is 10,000,000 times that of copper, a
result in accordance with experience.

Finally, by establishing a necessary relation between the motions of trans-
lation and rotation of all particles not spherical, we proved that a system of
such particles could not possibly satisfy the known relation between the two
specific heats of all gases.



[From the Philosophical Transactions, MDCCCLX.]



XXI. On the Theory of Compound Colours, and the Relations of the Colours
of the Spectrum. Communicated by Professor Stokes, Sec. R.S.

(Received January 5,— Read March 22, 1860.)



§ I. Introduction.

According to Newton's analysis of light ^', every colour in nature is pro-
duced by the mixture, in various proportions, of the different kinds of light
into which white %ht is divided by refraction. By means of a prism we may
analyse any coloured light, and determine the proportions in which the different
homogeneous rays enter into it; and by means of a lens we may recombine
these rays, and reproduce the original coloured light.

Newton has also shewnt how to combine the different rays of the spectrum
80 as to form a single beam of light, and how to alter the proportions of the
different colours so as to exhibit the result of combining them in any arbitrary
manner.

The number of different kinds of homogeneous light being infinite, and the
proportion in which each may be combined being also variable indefinitely, the
results of such combinations could not be appreciated by the eye, unless the
chromatic effect of every mixture, however complicated, could be expressed in
some simpler form. Colours, as seen by the human eye of the normal type, can
all be reduced to a few classes, and expressed by a few well-known names; and
even those colours which have different names have obvious relations among them-
selves. Every colour, except purple, is similar to some colour of the spectrum |,

* Optics, Book I. Part 2, Prop. 7.

t Lectiones Opticce, Part 2, § 1, pp. 100 to 105; and Optics, Book i. Part 2, Prop. 11.

X Optica, Book L Part 2, Prop. 4.



ON THE THEORY OF COMPOUND COLOURS. 411

although less intense ; and all purples may be compounded of blue and red,
and diluted with white to any required tint. Brown colours, which at first
sight seem different, are merely red, orange or yellow of feeble intensity, more
or less diluted with white.

It appears therefore that the result of any mixture of colours, however
complicated, may be defined by its relation to a certain small number of
well-known colours. Having selected our standard colours, and determined the
relations of a given colour to these, we have defined that colour completely as
to its appearance. Any colour which has the same relation to the standard
colours, will be identical in appearance, though its optical constitution, as
revealed by the prism, may be very different.

We may express this by saying that two compound colours may be chro-
matically identical, but optically different. The optical properties of light are
those which have reference to its origin and propagation through media, till it
falls on the sensitive organ of vision; the chromatical properties of light are
those which have reference to its power of exciting certain sensations of colour,
perceived through the organ of vision.

The investigation of the chromatic relations of the rays of the spectrum
must therefore be founded upon observations of the apparent identity of com-
pound colours, as seen by an eye either of the normal or of some abnormal
type; and the results to which the investigation leads must be regarded as
partaking of a physiological, as well as of a physical character, and as indicating
certain laws of sensation, depending on the constitution of the organ of vision,
which may be different in different individuals. We have to determine the
laws of the composition of colours in general, to reduce the number of standard
colours to the smallest possible, to discover, if we can, what they are, and to
ascertain the relation which the homogeneous light of different parts of the
spectrum bears to the standard colours.

§ II. History of the Theory of Compound Colours.

The foundation of the theory of the composition of colours was laid by
Newton*. He first shews that, by the mixture of homogeneal light, colours
may be produced which are "like to the colours of homogeneal light as to
the appearance of colour, but not as to the immutabOity of colour and consti-

* Optics, Book I. Part 2, Props. 4, 5, 6.



412 ON THE THEORY OF COMPOUND COLOURS.

tution of light." Red and yellow give an orange colour, which is chromatically
similar to the orange of the spectrum, but optically different, because it is
resolved into its component colours by a prism, while the orange of the spectrum
remains unchanged. When the colours to be mixed lie at a distance from one
another in the spectrum, the resultant appears paler than that intermediate
colour of the spectrum which it most resembles; and when several are mixed,
the resultant may appear white. Newton* is always careful, however, not to
call any mixture white, unless it agrees with comnon white light in its optical
as well as its chromatical properties, and is a mixture of all the homogeneal
colours. The theory of compound colours is first presented in a mathematical
form in Prop. 6, " In a mixture of priinary colours, the quantity arid quality
of each being given, to know the colour of the compound." He divides the
circumference of a circle into seven parts, proportional to the seven musical
intervals, in accordance with his opinion about the proportions of the colours
in the spectrum. At the centre of gravity of each of these arcs he places a
little circle, whose area is proportional to the number of rays of the corre-
sponding colour which enter into the given mixture. The position of the centre
of gravity of all these circles indicates the nature of the resultant colour. A
radius drawn through it points out that colour of the spectrum which it most
resembles, and the distance from the centre determines the fulness of its colour.

With respect to this construction, Newton says, " This rule I conceive
accurate enough for practice, though not mathematically accurate." He gives no
reasons for the different parts of his rule, but we shall find that his method
of finding the centre of gravity of the component colours is completely con-
firmed by my observations, and that it involves mathematically the theory of three
elements of colour ; but that the disposition of the colours on the circumference
of a circle was only a provisional arrangement, and that the true relations of
the colours of the spectrum can only be determined by direct observation.

Young t appears to have originated the theory, that the three elements of
colour are determined as much by the constitution of the sense of sight as by
anything external to us. He conceives that three different sensations may be
excited by light, but that the proportion in which each of the three is excited
depends on the nature of the light. He conjectures that these primary sensa-

* 7th and 8th Letters to Oldenburg.

+ Young's Lectures on Natural Philosophy, Kelland's Edition, p. 345, or Quarto, 1807, Vol. i.
p. 441 ; see also Young in Philosophical Transaction, 1801, or Works in Quarto, Vol il. p. 617.



ON THE THEORY OF COMPOUND COLOURS. 413

tions correspond to red, green, and violet. A blue ray, for example, though
homogeneous in itself, he conceives capable of exciting both the green and the
violet sensation, and therefore he would call blue a compound colour, though
the colour of a simple kind of light. The quality of any colour depends,
according to this theory, on the ratios of the intensities of the three sensations
which it excites, and its bHghtness depends on the sum of these three intensities.
Sir David Brewster, in his paper entitled " On a New Analysis of Solar
Light, indicating three Primary Colours, forming Coincident Spectra of equal
length*," regards the actual colours of the spectrum as arising from the inter-
mixture, in various proportions, of three primary kinds of light, red, yellow,
and blue, each of which is variable in intensity, but uniform in colour, from
one end of the spectrum to the other ; so that every colour in the spectrum
is really compound, and might be shewn to be so if we had the means of
separating its elements.

Sir David Brewster, in his researches, employed coloured media, which,
according to him, absorb the three elements of a single prismatic colour in
different degrees, and change their proportions, so as to alter the colour of the
light, without altering its refrangibility.

In this paper I shall not enter into the very important questions affecting
the physical theory of light, which can only be settled by a careful inquiry
into the phenomena of absorption. The physiological facts, that we have a
threefold sensation of colour, and that the three elements of this sensation are
affected in different proportions by light of different refrangibilities, are equally
true, whether we adopt the physical theory that there are three kinds of light
corresponding to these three colour-sensations, or whether we regard light of
definite refrangibility as an undulation of known length, and therefore variable
only in intensity, but capable of producing difierent chemical actions on different
substances, of being absorbed in different degrees by different media, and of
exciting in different degrees the three different colour-sensations of the human
eye.

Sir David Brewster has given a diagram of three curves, in which the
base-line represents the length of the spectrum, and the ordinates of the curves
represent, by estimation, the intensities of the three kinds of light at each point
of the spectrum. I have employed a diagram of the same kind to express the

* Transactions of the Royal Society of Edivimrgh, Vol. xii. p. 123.



414 ON THE THEORY OF COMPOUND COLOURS.

results arrived at in this paper, the ordinates being made to represent the
intensities of each of the three elements of colour, as calculated from the
experiments.

The most complete series of experiments on the mixture of the colours of
the spectrum, is that of Professor Helmholtz*, of Konlgsberg. By using two
sHts at right angles to one another, he formed two pure spectra, the fixed
lines of which were seen crossing one another when viewed in the ordinary-
way by means of a telescope. The colours of these spectra were thus combined
in every possible way, and the effect of the combination of any two could be
seen separately by drawing the eye back from the eye-piece of the telescope^
when the compound colour was seen by itself at the eye-hole. The proportion
of the components was altered by turning the combined slits round in their
own plane.

One result of these experiments was, that a colour, chromatically identical
with white, could be formed by combining yellow with indigo. M. Helmholtz.
was not then able to produce white with any other pair of simple colours, and
considered that three simple colours were required in general to produce white^
one from each of the three portions into which the spectrum is divided by
the yellow and indigo.

Professor Grassmannf shewed that Newton's theory of compound colours
implies that there are an infinite number of pairs of complementary colours in
the spectrum, and pointed out the means of finding them. He also shewed
how colours may be represented by lines, and combined by the method of the
parallelogram.

In a second memoirj, M. Helmholtz describes his method of ascertaining
these pairs of complementary colours. He formed a pure spectrum by means
of a slit, a prism, and a lens ; and in this spectrum he placed an apparatus
having two parallel slits which were capable of adjustment both in position
and breadth, so as to let through any two portions of the spectrum, in any
proportions. Behind this slit, these rays were united in an image of the prism,
which was received on paper. By arranging the slits, the colour of this image
may be reduced to white, and made identical with that of paper illuminated with
white light. The wave-lengths of the component colours were then measured by
observing the angle of diffraction through a grating. It was found that the

* Poggendorffs Anncden, Band lxxxvil {Philosophical Magazine, 1852, December).

t Ibid. Band lxxxix. (Philosophical Magazine, 1854, April). J Ibid. Band xciv.



ON THE THEORY OF COMPOUND COLOURS. 415

colours from red to green-yellow (X=2082) were complementary to colours ranging
from green-blue (X=1818) to violet, and that the colours between green-yellow
and green-blue have no homogeneous complementaries, but must be neutralized
by mixtures of red and violet.

M. Helmholtz also gives a provisional diagram of the curve formed by the
spectrum on Newton's diagram, for which his experiments did not furnish him
with the complete data.

Accounts of experiments by myself on the mixture of artificial colours by
rapid rotation, may be found in the Transactions of the Royal Society of
Edinburgh, Vol. xxi. Pt. 2 (1855); in an appendix to Professor George Wilson's
work on Coloiu - Blindness ; in the Report of the British Association for 1856,
p. 12; and in the Philosophical Magazine, July 1857, p. 40. These experiments
shew that, for the normal eye, there are three, and only three, elements of
colour, and that in the colour-blind one of these is absent. They also prove
that chromatic observations may be made, both by normal and abnormal eyes,
with such accuracy, as to warrant the employment of the results in the calcu-
lation of colour-equations, and in laying down colour-diagrams by Newton's rule.

The first instrument which I made (in 1852) to examine the mixtures of
the colours of the spectrum was similar to that which I now use, but smaller,
and it had no constant light for a term of comparison. The second was 6^ feet
long, made in 1855, and shewed tico combinations of colour side by side. I have
now succeeded in making the mixture much more perfect, and the comparisons
more exact, by using white reflected light, instead of the second compound
colour. An apparatus in which the light passes through the prisms, and is
reflected back again in nearly the same path by a concave mirror, was shewn
by me to the British Association in 1856. It has the advantage of being
portable, and need not be more than half the length of the other, in order
to produce a spectrum of equal length. I am so well satisfied with the working
of this form of the instrument, that I intend to make use of it in obtaining
equations from a gi-eater variety of observers than I could meet with when I
was obliged to use the more bulky instrument. It is difficult at first to get
the observer to believe that the compound light can ever be so adjusted as to
appear to his eyes identical with the white light in contact with it. He has to
learn what adjustments are necessary to produce the requisite alteration under
all circumstances, and he must never be satisfied till the two parts of the



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