[Footnote: Meeting of the British Association, Montreal. 1884. Section A.
Mathematical and Physical science. Opening Address by Prof. Sir William
Thomson, M.A., LL.D., D.C.L., F.R.SS.L. and E., F.R.A.S., President of the
Section.]
By Sir WILLIAM THOMSON.
The now well known kinetic theory of gases is a step so important in the
way of explaining seemingly static properties of matter by motion, that it
is scarcely possible to help anticipating in idea the arrival at a complete
theory of matter, in which all its properties will be seen to be merely
attributes of motion. If we are to look for the origin of this idea we must
go back to Democritus, Epicurus, and Lucretius. We may then, I believe,
without missing a single step, skip 1800 years. Early last century we find
in Malebranche's "Recherche de la Verite," the statement that "la durete de
corps" depends on "petits tourbillons." [1] These words, embedded in a
hopeless mass of unintelligible statements of the physical, metaphysical,
and theological philosophies of the day, and unsupported by any
explanation, elucidation, or illustration throughout the rest of the three
volumes, and only marred by any other single sentence or word to be found
in the great book, still do express a distinct conception which forms a
most remarkable step toward the kinetic theory of matter. A little later we
have Daniel Bernoulli's promulgation of what we now accept as a surest
article of scientific faith--the kinetic theory of gases. He, so far as I
know, thought only of Boyle's and Mariotte's law of the "spring of air," as
Boyle called it, without reference to change of temperature or the
augmentation of its pressure if not allowed to expand for elevation of
temperature, a phenomenon which perhaps he scarcely knew, still less the
elevation of temperature produced by compression, and the lowering of
temperature by dilatation, and the consequent necessity of waiting for a
fraction of a second or a few seconds of time (with apparatus of ordinary
experimental magnitude), to see a subsidence from a larger change of
pressure down to the amount of change that verifies Boyle's law. The
consideration of these phenomena forty years ago by Joule, in connection
with Bernoulli's original conception, formed the foundation of the kinetic
theory of gases as we now have it. But what a splendid and useful building
has been placed on this foundation by Clausius and Maxwell, and what a
beautiful ornament we see on the top of it in the radiometer of Crookes,
securely attached to it by the happy discovery of Tait and Dewar,[2] that
the length of the free path of the residual molecules of air in a good
modern vacuum may amount to several inches! Clausius' and Maxwell's
explanations of the diffusion of gases, and of thermal conduction in gases,
their charmingly intelligible conclusion that in gases the diffusion of
heat is just a little more rapid than the diffusion of molecules, because
of the interchange of energy in collisions between molecules,[3] while the
chief transference of heat is by actual transport of the molecules
themselves, and Maxwell's explanation of the viscosity of gases, with the
absolute numerical relations which the work of those two great discoverers
found among the three properties of diffusion, thermal conduction, and
viscosity, have annexed to the domain of science a vast and ever growing
province.
[Footnote 1: "Preuve de la supposition que j'ay faite: Que la matiere
subtile ou etheree est necessairement composee de PETITS TOURBILLONS; et
qu'ils sont les causes naturelles de tous les changements qui arrivent a la
matiere; ce que je confirme par i'explication des effets les plus generaux
de la Physique, tels que sont la durete des corps, leur fluidite, leur
pesanteur, legerete, la lumiere et la refraction et reflexion de ses
rayons."--Malebranche, "Recherche de la Verite," 1712.]
[Footnote 2: Proc. R.S.E., March 2, 1874, and July 5, 1875.]
[Footnote 3: On the other hand, in liquids, on account of the crowdedness
of the molecules, the diffusion of heat must be chiefly by interchange of
energies between the molecules, and should be, as experiment proves it is,
enormously more rapid than the diffusion of the molecules themselves, and
this again ought to be much less rapid than either the material or thermal
diffusivities of gases. Thus the diffusivity of common salt through water
was found by Fick to be as small as 0.0000112 square centimeter per second;
nearly 200 times as great as this is the diffusivity of heat through water,
which was found by J.T. Bottomley to be about 0.002 square centimeter per
second. The material diffusivities of gases, according to Loschmidt's
experiments, range from 0.98 (the interdiffusivity of carbonic acid and
nitrous oxide) to 0.642 (the interdiffusivity of carbonic oxide and
hydrogen), while the thermal diffusivities of gases, calculated according
to Clausius' and Maxwell's kinetic theory of gases, are 0.089 for carbonic
acid, 0.16 for common air of other gases of nearly the same density, and
1.12 for hydrogen (all, both material and thermal, being reckoned in square
centimeters per second).]
Rich as it is in practical results, the kinetic theory of gases, as
hitherto developed, stops absolutely short at the atom or molecule, and
gives not even a suggestion toward explaining the properties in virtue of
which the atoms or molecules mutually influence one another. For some
guidance toward a deeper and more comprehensive theory of matter, we may
look back with advantage to the end of last century and beginning of this
century, and find Rumford's conclusion regarding the heat generated in
boring a brass gun: "It appears to me to be extremely difficult, if not
quite impossible, to form any distinct idea of anything capable of being
excited and communicated in the manner the heat was excited and
communicated in these experiments, except it be MOTION;" and Davy's still
more suggestive statements: "The phenomena of repulsion are not dependent
on a peculiar elastic fluid for their existence." ... "Heat may be defined
as a peculiar motion, probably a vibration, of the corpuscles of bodies,
tending to separate them." ... "To distinguish this motion from others, and
to signify the causes of our sensations of heat, etc., the name _repulsive_
motion has been adopted." Here we have a most important idea. It would be
somewhat a bold figure of speech to say the earth and moon are kept apart
by a repulsive motion; and yet, after all, what is centrifugal force but a
repulsive motion, and may it not be that there is no such thing as
repulsion, and that it is solely by inertia that what seems to be repulsion
is produced? Two bodies fly together, and, accelerated by mutual
attraction, if they do not precisely hit one another, they cannot but
separate in virtue of the inertia of their masses. So, after dashing past
one another in sharply concave curves round their common center of gravity,
they fly asunder again. A careless onlooker might imagine they had repelled
one another, and might not notice the difference between what he actually
sees and what he would see if the two bodies had been projected with great
velocity toward one another, and either colliding and rebounding, or
repelling one another into sharply convex continuous curves, fly asunder
again.
Joule, Clausius, and Maxwell, and no doubt Daniel Bernoulli himself, and I
believe every one who has hitherto written or done anything very explicit
in the kinetic theory of gases, has taken the mutual action of molecules in
collision as repulsive. May it not after all be attractive? This idea has
never left my mind since I first read Davy's "Repulsive Motion," about
thirty-five years ago, and I never made anything of it, at all events have
not done so until to-day (June 16, 1884)--if this can be said to be making
anything of it--when, in endeavoring to prepare the present address, I
notice that Joule's and my own old experiments[1] on the thermal effect of
gases expanding from a high-pressure vessel through a porous plug, proves
the less dense gas to have greater intrinsic _potential_ energy than the
denser gas, if we assume the ordinary hypothesis regarding the temperature
of a gas, according to which two gases are of equal temperatures [2] when
the kinetic energies of their constituent molecules are of equal average
amounts per molecule.
[Footnote 1: Republished in Sir W. Thomson's "Mathematical and Physical
Papers," vol. i., article xlix., p. 381. ]
[Footnote 2: That this is a mere hypothesis has been scarcely remarked by
the founders themselves, nor by almost any writer on the kinetic theory of
gases. No one has yet examined the question, What is the condition as
regards average distribution of kinetic energy, which is ultimately
fulfilled by two portions of gaseous matter, separated by a thin elastic
septum which absolutely prevents interdiffusion of matter, while it allows
interchange of kinetic energy by collisions against itself? Indeed, I do
not know but, that the present is the very first statement which has ever
been published of this condition of the problem of equal temperatures
between two gaseous masses.]
Think of the thing thus. Imagine a great multitude of particles inclosed by
a boundary which may be pushed inward in any part all round at pleasure.
Now station an engineer corps of Maxwell's army of sorting demons all round
the inclosure, with orders to push in the boundary diligently everywhere,
when none of the besieged troops are near, and to do nothing when any of
them are seen approaching, and until after they have turned again inward.
The result will be that, with exactly the same sum of kinetic and potential
energies of the same inclosed multitude of particles, the throng has been
caused to be denser. Now Joule's and my own old experiments on the efflux
of air prove that if the crowd be common air, or oxygen, or nitrogen, or
carbonic acid, the temperature is a little higher in the denser than in the
rarer condition when the energies are the same. By the hypothesis, equality
of temperature between two different gases or two portions of the same gas
at different densities means equality of kinetic energies in the same
number of molecules of the two. From our observations proving the
temperature to be higher, it therefore follows that the potential energy is
smaller in the condensed crowd. This--always, however, under protest as to
the temperature hypothesis--proves some degree of attraction among the
molecules, but it does not prove ultimate attraction between two molecules
in collision, or at distances much less than the average mutual distance of
nearest neighbors in the multitude. The collisional force might be
repulsive, as generally supposed hitherto, and yet attraction might
predominate in the whole reckoning of difference between the intrinsic
potential energies of the more dense and less dense multitudes.
It is however remarkable that the explanation of the propagation of sound
through gases, and even of the positive fluid pressure of a gas against the
sides of the containing vessel, according to the kinetic theory of gases,
is quite independent of the question whether the ultimate collisional force
is attractive or repulsive. Of course it must be understood that, if it is
attractive, the particles must, be so small that they hardly ever
meet--they would have to be infinitely small to _never_ meet--that, in
fact, they meet so seldom, in comparison with the number of times their
courses--are turned through large angles by attraction, that the influence
of these surely attractive collisions is preponderant over that of the
comparatively very rare impacts from actual contact. Thus, after all, the
train of speculation suggested by Davy's "Repulsive Motion" does not allow
us to escape from the idea of true repulsion, does not do more than let us
say it is of no consequence, nor even say this with truth, because, if
there are impacts at all, the nature of the force during the impact and the
effects of the mutual impacts, however rare, cannot be evaded in any
attempt to realize a conception of the kinetic theory of gases. And in
fact, unless we are satisfied to imagine the atoms of a gas as mathematical
points endowed with inertia, and as, according to Boscovich, endowed with
forces of mutual, positive, and negative attraction, varying according to
some definite function of the distance, we cannot avoid the question of
impacts, and of vibrations and rotations of the molecules resulting from
impacts, and we must look distinctly on each molecule as being either a
little elastic solid or a configuration of motion in a continuous
all-pervading liquid. I do not myself see how we can ever permanently rest
anywhere short of this last view; but it would be a very pleasant temporary
resting-place on the way to it if we could, as it were, make a mechanical
model of a gas out of little pieces of round, perfectly elastic solid
matter, flying about through the space occupied by the gas, and colliding
with one another and against the sides of the containing vessel.
This is, in fact, all we have of the kinetic theory of gases up to the
present time, and this has done for us, in the hands of Clausius and
Maxwell, the great things which constitute our first step toward a
molecular theory of matter. Of course from it we should have to go on to
find an explanation of the elasticity and all the other properties of the
molecules themselves, a subject vastly more complex and difficult than the
gaseous properties, for the explanation of which we assume the elastic
molecule; but without any explanation of the properties of the molecule
itself, with merely the assumption that the molecule has the requisite
properties, we might rest happy for a while in the contemplation of the
kinetic theory of gases, and its explanation of the gaseous properties,
which is not only stupendously important as a step toward a more
thoroughgoing theory of matter, but is undoubtedly the expression of a
perfectly intelligible and definite set of facts in Nature.
But alas for our mechanical model consisting of the cloud of little elastic
solids flying about among one another. Though each particle have absolutely
perfect elasticity, the end must be pretty much the same as if it were but
imperfectly elastic. The average effect of repeated and repeated mutual
collisions must be to gradually convert all the translational energy into
energy of shriller and shriller vibrations of the molecule. It seems
certain that each collision must have something more of energy in
vibrations of very finely divided nodal parts than there was of energy in
such vibrations before the impact. The more minute this nodal subdivision,
the less must be the tendency to give up part of the vibrational energy
into the shape of translational energy in the course of a collision; and I
think it is rigorously demonstrable that the whole translational energy
must ultimately become transformed into vibrational energy of higher and
higher nodal subdivisions if each molecule is a continuous elastic solid.
Let us, then, leave the kinetic theory of gases for a time with this
difficulty unsolved, in the hope that we or others after us may return to
it, armed with more knowledge of the properties of matter, and with sharper
mathematical weapons to cut through the barrier which at present hides from
us any view of the molecule itself, and of the effects other than mere
change of translational motion which it experiences in collision.
To explain the elasticity of a gas was the primary object of the kinetic
theory of gases. This object is only attainable by the assumption of an
elasticity more complex in character, and more difficult of explanation,
than the elasticity of gases--the elasticity of a solid. Thus, even if the
fatal fault in the theory, to which I have alluded, did not exist, and if
we could be perfectly satisfied with the kinetic theory of gases founded on
the collisions of elastic solid molecules, there would still be beyond it a
grander theory which need not be considered a chimerical object of
scientific ambition--to explain the elasticity of solids. But we may be
stopped when we commence to look in the direction of such a theory with the
cynical question, What do you mean by explaining a property of matter? As
to being stopped by any such question, all I can say is that if engineering
were to be all and to end all physical science, we should perforce be
content with merely finding properties of matter by observation, and using
them for practical purposes. But I am sure very few, if any, engineers are
practically satisfied with so narrow a view of their noble profession. They
must and do patiently observe, and discover by observation, properties of
matter and results of material combinations. But deeper questions are
always present, and always fraught with interest to the true engineer, and
he will be the last to give weight to any other objection to any attempt to
see below the surface of things than the practical question, Is it likely
to prove wholly futile? But now, instead of imagining the question, What do
you mean by explaining a property of matter? to be put cynically, and
letting ourselves be irritated by it, suppose we give to the questioner
credit for being sympathetic, and condescend to try and answer his
question. We find it not very easy to do so. All the properties of matter
are so connected that we can scarcely imagine one _thoroughly explained_
without our seeing its relation to all the others, without in fact having
the explanation of all; and till we have this we cannot tell what we mean
by "explaining a property" or "explaining the properties" of matter. But
though this consummation may never be reached by man, the progress of
science may be, I believe will be, step by step toward it, on many
different roads converging toward it from all sides. The kinetic theory of
gases is, as I have said, a true step on one of the roads. On the very
distinct road of chemical science, St. Claire Deville arrived at his grand
theory of dissociation without the slightest aid from the kinetic theory of
gases. The fact that he worked it out solely from chemical observation and
experiment, and expounded it to the world without any hypothesis whatever,
and seemingly even without consciousness of the beautiful explanation it
has in the kinetic theory of gases, secured for it immediately an
independent solidity and importance as a chemical theory when he first
promulgated it, to which it might even by this time scarcely have attained
if it had first been suggested as a probability indicated by the kinetic
theory of gases, and been only afterward confirmed by observation. Now,
however, guided by the views which Clausius and Williamson have given us of
the continuous interchange of partners between the compound molecules
constituting chemical compounds in the gaseous state, we see in Deville's
theory of dissociation a point of contact of the most transcendent interest
between the chemical and physical lines of scientific progress.
To return to elasticity: if we could make out of matter devoid of
elasticity a combined system of relatively moving parts which, in virtue of
motion, has the essential characteristics of an elastic body, this would
surely be, if not positively a step in the kinetic theory of matter, at
least a fingerpost pointing a way which we may hope will lead to a kinetic
theory of matter. Now this, as I have already shown,[1] we can do in
several ways. In the case of the last of the communications referred to, of
which only the title has hitherto been published, I showed that, from the
mathematical investigation of a gyrostatically dominated combination
contained in the passage of Thomson and Tait's "Natural Philosophy"
referred to, it follows that any ideal system of material particles, acting
on one another mutually through massless connecting springs, may be
perfectly imitated in a model consisting of rigid links jointed together,
and having rapidly rotating fly wheels pivoted on some or on all of the
links. The imitation is not confined to cases of equilibrium. It holds also
for vibration produced by disturbing the system infinitesimally from a
position of stable equilibrium and leaving it to itself. Thus we may make a
gyrostatic system such that it is in equilibrium under the influence of
certain positive forces applied to different points of this system; all the
forces being precisely the same as, and the points of application similarly
situated to, those of the stable system with springs. Then, provided proper
masses (that is to say, proper amounts and distributions of inertia) be
attributed to the links, we may remove the external forces from each
system, and the consequent vibration of the points of application of the
forces will be identical. Or we may act upon the systems of material points
and springs with any given forces for any given time, and leave it to
itself, and do the same thing for the gyrostatic system; the consequent
motion will be the same in the two cases. If in the one case the springs
are made more and more stiff, and in the other case the angular velocities
of the fly wheels are made greater and greater, the periods of the
vibrational constituents of the motion will become shorter and shorter, and
the amplitudes smaller and smaller, and the motions will approach more and
more nearly those of two perfectly rigid groups of material points moving
through space and rotating according to the well known mode of rotation of
a rigid body having unequal moments of inertia about its three principal
axes. In one case the ideal nearly rigid connection between the particles
is produced by massless, exceedingly stiff springs; in the other case it is
produced by the exceedingly rapid rotation of the fly wheels in a system
which, when the fly wheels are deprived of their rotation, is perfectly
limp.
[Footnote 1: Paper on "Vortex Atoms," _Proc_. R.S.E. February. 1867:
abstract of a lecture before the Royal Institution of Great Britain, March
4, 1881, on "Elasticity Viewed as possibly a Mode of Motion"; Thomson and
Tait's "Natural Philosophy," second edition, part 1, §§ 345 viii. to 345
xxxvii.; "On Oscillation and Waves in an Adynamic Gyrostatic System" (title
only), _Proc_. R.S.E. March, 1883.]
The drawings (Figs. 1 and 2) before you illustrate two such material
systems.[1] The directions of rotation of the fly-wheels in the gyrostatic
system (Fig. 2) are indicated by directional ellipses, which show in
perspective the direction of rotation of the fly-wheel of each gyrostat.
The gyrostatic system (Fig. 2) might have been constituted of two
gyrostatic members, but four are shown for symmetry. The inclosing circle
represents in each case in section an inclosing spherical shell to prevent
the interior from being seen. In the inside of one there are fly-wheels, in
the inside of the other a massless spring. The projecting hooked rods seem
as if they are connected by a spring in each case. If we hang any one of
the systems up by the hook on one of its projecting rods, and hang a weight
to the hook of the other projecting rod, the weight, when first put on,
will oscillate up and down, and will go on doing so for ever if the system
be absolutely unfrictional. If we check the vibration by hand, the weight
will hang down at rest, the pin drawn out to a certain degree; and the
distance drawn out will be simply proportional to the weight hung on, as in
an ordinary spring balance.
[Footnote 1: In Fig. 1 the two hooked rods seen projecting from the sphere
are connected by an elastic coach-spring. In Fig. 2 the hooked rods are
connected one to each of two opposite corners of a four-sided jointed
frame, each member of which carries a gyrostat so that the axis of rotation
of the fly-wheel is in the axis of the member of the frame which bears it.
Each of the hooked rods in Fig. 2 is connected to the framework through a
swivel joint, so that the whole gyrostatic framework may be rotated about
the axis of the hooked rods in order to annul the moment of momentum of the
framework about this axis due to rotation of the fly-wheels in the
gyrostat.]
[Illustration: FIG. 1]
[Illustration: FIG. 2]
Here, then, out of matter possessing rigidity, but absolutely devoid of
elasticity, we have made a perfect model of a spring in the form of a
spring balance. Connect millions of millions of particles by pairs of rods
such as these of this spring balance, and we have a group of particles
constituting an elastic solid; exactly fulfilling the mathematical ideal
worked out by Navier, Poisson, and Cauchy, and many other mathematicians,
who, following their example, have endeavored to found a theory of the
elasticity of solids on mutual attraction and repulsion between a group of
material particles. All that can possibly be done by this theory, with its
assumption of forces acting according to any assumed law of relation to
distance, is done by the gyrostatic system. But the gyrostatic system does,
besides, what the system of naturally acting material particles cannot
do--it constitutes an elastic solid which can have the Faraday
magneto-optic rotation of the plane of polarization of light; supposing the
application of our solid to be a model of the luminiferous ether for
illustrating the undulatory theory of light. The gyrostatic model spring
balance is arranged to have zero moment of momentum as a whole, and
therefore to contribute nothing to the Faraday rotation; with this
arrangement the model illustrates the luminiferous ether in a field
unaffected by magnetic force. But now let there be a different rotational
velocity imparted to the jointed square round the axis of the two
projecting hooked rods, such as to give a resultant moment of momentum
round any given line through the center of inertia of the system; and let
pairs of the hooked rods in the model thus altered, which is no longer a
model of a mere spring balance, be applied as connections between millions
of pairs of particles as before, with the lines of resultant moment of
momentum all similarly directed. We now have a model elastic solid which
will have the property that the direction of vibration in waves of
rectilinear vibrations propagated through it shall turn round the line of
propagation of the waves, just as Faraday's observation proves to be done
by the line of vibration of light in a dense medium between the poles of a
powerful magnet. The case of wave front perpendicular to the lines of
resultant moment of momentum (that is to say, the direction of propagation
being parallel to these lines) corresponds, in our mechanical model, to the
case of light traveling in the direction of the lines of force in a
magnetic field.
In these illustrations and models we have different portions of ideal rigid
matter acting upon one another, by normal pressure at mathematical points
of contact--of course no forces of friction are supposed. It is exceedingly
interesting to see how thus, with no other postulates than inertia,
rigidity, and mutual impenetrability, we can thoroughly model not only an
elastic solid, and any combination of elastic solids, but so complex and
recondite a phenomenon as the passage of polarized light through a magnetic
field. But now, with the view of ultimately discarding the postulate of
rigidity from all our materials, let us suppose some to be absolutely
destitute of rigidity, and to possess merely inertia and incompressibility,
and mutual impenetrability with reference to the still remaining rigid
matter. With these postulates we can produce a perfect model of mutual
action at a distance between solid particles, fulfilling the condition, so
keenly desired by Newton and Faraday, of being explained by continuous
action through an intervening medium. The law of the mutual force in our
model, however, is not the simple Newtonian law, but the much more complex
law of the mutual action between electro magnets--with this difference,
that in the hydro-kinetic model in every case the force is opposite in
direction to the corresponding force in the electro-magnetic analogue.
Imagine a solid bored through with a hole, and placed in our ideal perfect
liquid. For a moment let the hole be stopped by a diaphragm, and let an
impulsure pressure be applied for an instant uniformly over the whole
membrane, and then instantly let the membrane be dissolved into liquid.
This action originates a motion of the liquid relatively to the solid, of a
kind to which I have given the name of "irrotational circulation," which
remains absolutely constant however the solid be moved through the liquid.
Thus, at any time the actual motion of the liquid at any point in the
neighborhood of the solid will be the resultant of the motion it would have
in virtue of the circulation alone, were the solid at rest, and the motion
it would have in virtue of the motion of the solid itself, had there been
no circulation established through the aperture. It is interesting and
important to remark in passing that the whole kinetic energy of the liquid
is the sum of the kinetic energies which it would have in the two cases
separately. Now, imagine the whole liquid to be inclosed in an infinitely
large, rigid, containing vessel, and in the liquid, at an infinite distance
from any part of the containing vessel, let two perforated solids, with
irrotational circulation through each, be placed at rest near one another.
The resultant fluid motion due to the two circulations, will give rise to
fluid pressure on the two bodies, which, if unbalanced, will cause them to
move. The force systems--force-and-torques, or pairs of forces--required to
prevent them from moving will be mutual and opposite, and will be the same
as, but opposite in direction to, the mutual force systems required to hold
at rest two electromagnets fulfilling the following specification: The two
electro magnets are to be of the same shape and size as the two bodies, and
to be placed in the same relative positions, and to consist of infinitely
thin layers of electric currents in the surfaces of solids possessing
extreme diamagnetic quality--in other words, infinitely small permeability.
The distribution of electric current on each body may be any whatever which
fulfills the condition that the total current across any closed line drawn
on the surface once through the aperture is equal to ¼ [pi] of the
circulation[1] through the aperture in the hydro-kinetic analogue.
[Footnote 1: The integral of tangential component velocity all round any
closed curve, passing once through the aperture, is defined as the
"cyclic-constant" or the "circulation" ("Vortex Motion," § 60 (a), _Trans_.
R.S.E., April 29, 1867). It has the same value for all closed curves
passing just once through the aperture, and it remains constant through all
time, whether the solid body be in motion or at rest.]
It might be imagined that the action at a distance thus provided for by
fluid motion could serve as a foundation for a theory of the equilibrium,
and the vibrations, of elastic solids, and the transmission of waves like
those of light through an extended quasi-elastic solid medium. But
unfortunately for this idea the equilibrium is essentially unstable, both
in the case of magnets and, notwithstanding the fact that the forces are
oppositely directed, in the hydro-kinetic analogue also, when the several
movable bodies (two or any greater number) are so placed relatively as to
be in equilibrium. If, however, we connect the perforated bodies with
circulation through them in the hydro-kinetic system, by jointed rigid
connecting links, we may arrange for configurations of stable equilibrium.
Thus, without fly-wheels, but with fluid circulations through apertures, we
may make a model spring balance or a model luminiferous ether, either
without or with the rotational quality corresponding to that of the true
luminiferous ether in the magnetic fluid--in short, do all by the
perforated solids with circulations through them that we saw we could do by
means of linked gyrostats. But something that we cannot do by linked
gyrostats we can do by the perforated bodies with fluid circulation: we can
make a model gas. The mutual action at a distance, repulsive or attractive
according to the mutual aspect of the two bodies when passing within
collisional distance[1] of one another, suffices to produce the change of
direction of motion in collision, which essentially constitutes the
foundation of the kinetic theory of gases, and which, as we have seen
before, may as well be due to attraction as to repulsion, so far as we know
from any investigation hitherto made in this theory.
[Footnote 1: According to this view, there is no precise distance, or
definite condition respecting the distance, between two molecules, at which
apparently they come to be in collision, or when receding from one another
they cease to be in collision. It is convenient, however, in the kinetic
theory of gases, to adopt arbitrarily a precise definition of collision,
according to which two bodies or particles mutually acting at a distance
may be said to be in collision when their mutual action exceeds some
definite arbitrarily assigned limit, as, for example, when the radius of
curvature of the path of either body is less than a stated fraction (one
one-hundredth, for instance) of the distance between them.]
There remains, however, as we have seen before, the difficulty of providing
for the case of actual impacts between the solids, which must be done by
giving them massless spring buffers or, which amounts to the same thing,
attributing to them repulsive forces sufficiently powerful at very short
distances to absolutely prevent impacts between solid and solid; unless we
adopt the equally repugnant idea of infinitely small perforated solids,
with infinitely great fluid circulations through them. Were it not for this
fundamental difficulty, the hydro-kinetic model gas would be exceedingly
interesting; and, though we could scarcely adopt it as conceivably a true
representation of what gases really are, it might still have some
importance as a model configuration of solid and liquid matter, by which
without elasticity the elasticity of true gas might be represented.
But lastly, since the hydro-kinetic model gas with perforated solids and
fluid circulations through them fails because of the impacts between the
solids, let us annul the solids and leave the liquid performing
irrotational circulation round vacancy,[1] in the place of the solid cores
which we have hitherto supposed; or let us annul the rigidity of the solid
cores of the rings, and give them molecular rotation according to
Helmholtz's theory of vortex motion. For stability the molecular rotation
must be such as to give the same velocity at the boundary of the rotational
fluid core as that of the irrotationally circulating liquid in contact with
it, because, as I have proved, frictional slip between two portions of
liquid in contact is inconsistent with stability. There is a further
condition, upon which I cannot enter into detail just now, but which may be
understood in a general way when I say that it is a condition of either
uniform or of increasing molecular rotation from the surface inward,
analogous to the condition that the density of a liquid, resting for
example under the influence of gravity, must either be uniform or must be
greater below than above for stability of equilibrium. All that I have said
in favor of the model vortex gas composed of perforated solids with fluid
circulations through them holds without modification for the purely
hydro-kinetic model, composed of either Helmholtz cored vortex rings or of
coreless vortices, and we are now troubled with no such difficulty as that
of the impacts between solids. Whether, however, when the vortex theory of
gases is thoroughly worked out, it will or will not be found to fail in a
manner analogous to the failure which I have already pointed out in
connection with the kinetic theory of gases composed of little elastic
solid molecules, I cannot at present undertake to speak with certainty. It
seems to me most probable that the vortex theory cannot fail in any such
way, because all I have been able to find out hitherto regarding the
vibration of vortices,[2] whether cored or coreless, does not seem to imply
the liability of translational or impulsive energies of the individual
vortices becoming lost in energy of smaller and smaller vibrations.
[Footnote 1: Investigations respecting coreless vortices will be found in a
paper by the author, "Vibrations of a Columnar Vortex," _Proc_. R.S.E.,
March 1, 1880; and a paper by Hicks, recently read before the Royal
Society.]
[Footnote 2: See papers by the author "On Vortex Motion." _Trans_. R.S.E.
April, 1867, and "Vortex Statics," _Proc_. R.S.E. December, 1875; also a
paper by J.J. Thomson, B.A., "On the Vibrations of a Vortex Ring," _Trans_.
R.S. December, 1881, and his valuable book on "Vortex Motion."]
As a step toward kinetic theory of matter, it is certainly most interesting
to remark that in the quasi-elasticity, elasticity looking like that of an
India-rubber band, which we see in a vibrating smoke-ring launched from an
elliptic aperture, or in two smoke-rings which were circular, but which
have become deformed from circularity by mutual collision, we have in
reality a virtual elasticity in matter devoid of elasticity, and even
devoid of rigidity, the virtual elasticity being due to motion, and
generated by the generation of motion.
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