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heater as far as its frame would allow.

From the fact that the fibre of silk had shown positive motion so nearly
up to the pressure of the atmosphere it might have been anticipated that
the spider line, on account of its much greater thinness, would have shown
positive motion even at pressures considerably above that of the atmosphere.
But the reasoning of Art. 46 respecting the differences of temperature to be
maintained and the effect of the air currents, obviously applies with greater
force to the spider line than to the fibre of silk, and at once accounts for the
observed fact that the positive motion with the spider line was not obtained
until the pressures were somewhat lower than those necessary for the fibre
of silk.



304 ON CERTAIN DIMENSIONAL PROPERTIES OF MATTER [33

52. Both with the fibre of silk and the spider line, the phenomena of
impulsion (the excess of pressure against warm surfaces) were apparent and
consistent at densities many hundred times greater than the highest densities
at which like results are obtained with vanes several hundred times broader
than the fibre of silk ; this verifies the theoretical conclusion on which this
part of the investigation was based. The results in this case are not so
definite as is the agreement of the logarithmic homologues in the instances
of transpiration ; but the one fact supports the other, and we may consider
the law of impulsion Law VIII., Art. 9 to have been sufficiently proved.

This concludes the experimental investigation.



PART II. (THEOEETICAL).

SECTION V. INTRODUCTION TO THE THEORY.

53. In suggesting in a former paper that the results discovered by
Mr Crookes were due to the communication of heat from the surface of the
solid bodies to the gas surrounding them, I pointed out as the fundamental
fact on which I based my explanation, that when heat is communicated from
a solid surface to a gas, the mean velocity of the molecules which rebound
from the surface must be greater as they rebound than as they approach, and
hence the momentum which these particular molecules communicate to the
surface must be greater than it would be if the surface were at the same
temperature as the gas.

So far the reasoning is incontrovertible. But in order to explain the ex-
perimental results, it was necessary to assume that the number of cold
molecules which approached the hot surface would be the same as if the
surface were at the same temperature as the gas, or at any rate, if reduced,
the number would not be sufficiently reduced to counteract the effect of
increased velocity of rebound.

Although at that time I could not see any definite proof of this, nor any
way of definitely examining the question, yet I had a strong impression that
the assumption was legitimate ; and although I hoped at some future time
to be able to complete the theoretical explanation, I was content for the time
to rest the evidence of the truth of the assumptions involved on the adequacy
of the reasoning to explain the experimental results obtained.

As other suggestions respecting the cause of the phenomena, widely
different in character from mine, had found supporters, and a good deal of
scepticism was expressed as to the fitness of the cause which I had suggested,
my attention was occupied in deducing the actions which must result from
such a force, and comparing them with experimental ^results. Having, how-
ever, at length satisfied myself, and seeing that a conviction was spreading



33] IN THE GASEOUS STATE. 305

that what I suggested contained the germ of the explanation, I set to work
in earnest to complete the explanation, and ascertain by an extension of the
dynamical theory of gases what effect the hot molecules receding from
the surface should produce on the number arid temperature of those
approaching.

My first attempts to accomplish this were altogether unsuccessful. When
contemplating the phenomena it seemed to me that I could perceive a
glimmering of the method of reasoning for which I was in search, but as
soon as ever I attempted to give definite expression to it this glimmering
vanished.

The reason for this I now perceive clearly. When contemplating the
phenomena, I had a surface of limited extent before me, and I considered the
effect on such a surface without recognising the fundamental importance of
the limit to size.

On the other hand, when I came to definite reasoning, for the sake of
what appeared to be a simplification of the conditions of the problem, I
assumed the surface to be without limit, thus introducing a fundamental
alteration into the conditions of the problem without perceiving it.

The importance of this limit only became apparent to me when I found,
by simple dynamical reasoning, that with surfaces of unlimited extent such
results as those actually obtained would be impossible. This appeared as
follows :

No force on unlimited surface.

54. If we had two plane plates of unlimited extent, H and C, the surface
of H opposite to C being hotter than the surface of C which was opposite to
H, the outside surfaces of both plates being at the same temperature, then
in order to produce results similar to those obtained with limited plates, the
gas between the two plates must maintain a greater steady pressure on the
plate H, than that which it exerts on the colder plate C. Whereas it is at
once obvious that such a condition is contrary to the laws of motion, which
require that the gas between the two surfaces should exert an equal and
opposite pressure on both surfaces.

Having once perceived the force of this reasoning, it became clear to me
that if, as I had supposed, the results obtained in the experiments were due
to gaseous pressure, then they must depend on the limited extent of the
surfaces.

This gave me the clue, in following which I have not only had the satis-
faction of finding the explanation complete as regards the phenomena from
o. R. 20



306 ON CERTAIN DIMENSIONAL PROPERTIES OF MATTER [33

which it originated, but I have also found that the theory indicated the
phenomena of thermal transpiration, and explains much that hitherto has
been considered anomalous respecting the laws of transpiration of gases
through small channels suggesting the experiments by which might be
established the relation between these actions.

The manner in which the force arises in the case of a limited surface was
at first rendered much clearer to me by considering an illustration, which
I introduce here, although it forms no part of the proof which will follow.

55. Instead of H and C being plates with gas between them, let them
be earthen batteries of unlimited length, and suppose that guns are distri-
buted at uniform intervals along those batteries ; suppose, also, that all the
shot fired from H bury themselves in the earth of C, and vice versd.

Then, in the first place, it is obvious that since on firing a shot the
momentum imparted to the gun is equal and opposite to the momentum given
to the shot, every shot fired from H will exercise the same force to move the
battery H away from C as the shot will exercise to move C away from H ;
and in the same way the recoil of the guns on C will exercise the same
tendency to move C away from H as the shot will exercise to move H away
from G. And this will be the case whether the guns are supposed to be
pointed straight across the interval between the batteries, or, as / shall
suppose, are pointed with various degrees of obliquity.

Since, then, the result of every shot, whether fired from H or G, causes
equal and opposite forces on the two batteries, the result of all the firing, no
matter how much harder one battery may bombard than the other, must be
to cause an equal force on each battery, the batteries being of unlimited
length.

This case will be seen to be strictly analogous to the effect of the gas
between two plates of unlimited extent to cause equal pressures on the
plates, no matter what may be the differences in the temperature of the
plates.

If now we consider the batteries of limited extent, then, owing to the
obliquity of the guns, some of the shot from H may pass beyond the ends
of C, and vice versd ; and in this case the force of recoil on the battery which
fires will no longer be balanced by the stopping of the shot on the other
battery. So that supposing the directions of firing to be similar, that
battery which fires the hardest will be subject to the greatest tendency to
move back.

The battery which fires the hardest corresponds with the hottest plate ;
and hence we perceive by analogy that, if of limited extent, the hottest
plate will experience the greatest pressure from the gas between the plates.



33] IN THE GASEOUS STATE. 307

56. The analogy between the batteries and the plates is rendered more
strict if we suppose the batteries H and C to be two limited batteries, each
placed in front of a battery of unlimited extent, and that these unlimited
batteries are pounding away in an exactly similar manner.

The effect of the shot from these unlimited batteries on H and G will be
analogous to the effect of the gas outside and beyond the plates. And it is
at once seen that these unlimited batteries will produce similar effects on H
and G respectively, and that the effect of the firing between H and G will
be uninfluenced by the batteries behind, and therefore, as before, that battery
will be subject to the greatest tendency to move back which fires the hardest.

To make the analogy between the two cases complete, suppose that H
and G, in addition to pounding away at each other, are exactly returning the
fire of the batteries from behind, and that the mean rate at which H fires at
G and G at H are exactly the same as the rate at which the other firing goes
on, but that the velocity of the shot from H is just as much greater than
the mean velocity, as the velocity of the shot from G is below the mean.
Then it is at once seen that the total tendency on H is to move back, while
the total tendency on G is to move forward.

It obviously follows from the foregoing that the inequality in the forces
on H and C could only occur at a certain distance from their ends, which
distance would depend on the distance between the batteries ; and hence
that the ratio which this inequality (due to any particular rate of firing)
would bear to the whole reaction on either battery would increase as the
length of the batteries diminished ; or in other words, the inequality of force
would be proportional to the distance between the batteries, and would be
constant whatever might be the length of the batteries beyond a certain
point.

At first sight it may appear that the distance between the batteries H
and G should be analogous to the distance between the hot and cold plates ;
but it is necessary to remember that it is only in case of the gas being
extremely rare, as compared with the distance between the plates, that the
molecules can be supposed to go straight from the one plate to the other.
In ordinary cases the molecules encounter other molecules, and the effect of
such encounters is to reduce the motion to a mean. Hence it appears that
the distance between the batteries as affecting the equality in the reactions
is somewhat analogous to the distance which a molecule may be supposed to
travel without losing its characteristic motion. And hence it would appear
that in the case of gas the inequalities of force on the two plates would be
proportional to the inverse density of the gas and the extent of the boundaries
of plates.

57. The shot from H which miss G, and those from G which miss //,

202



308 ON CERTAIN DIMENSIONAL PROPERTIES OF MATTER [33

must be stopped by the outside batteries. Therefore the inequalities in the
forces on H and C will be balanced by inequalities in the forces on the
batteries behind, and the sum of the forces on // and the battery behind will
be equal to the sum of the forces on C and the battery behind.

And this is strictly analogous to the result of Schuster's experiment, viz. :
that the effect upon the vanes of the light mill is exactly balanced by the
effect on the containing vessel.

58. The batteries also serve to illustrate the action of thermal tran-
spiration. In the case already considered (Art. 57) the inequality between
the shot from H which miss C and those from C which miss H is transferred
to the outside batteries, or in the case of the gas. to the containing vessel.
The better to illustrate the present point, suppose that the outside batteries
are ranged across the ends of the open space between H and C, This will
make no difference to the result. The inequality of the action of the shot
which miss H and C must now cause a force parallel to the end batteries,
tending to cause these batteries to move end-wise in the direction of C.

Suppose that the two batteries H and C were free to move together in
the direction from C vo H (suppose them on a truck). The inequality in the
force would set them in motion in this direction, which motion would increase
until the actual velocity of the shot from C equalled the actual velocity of
the shot from H; then all inequalities in the reactions would cease, and there
would be no reactions on the limiting batteries.

In this case the limiting batteries are obviously analogous to the sides of
a tube, and the interval between the planes H and C corresponds with a layer
of gas at equal pressures, but across which the heat is being conducted by the
greater velocity of the molecules which move from H to C ; and the con-
clusion is that such a layer of gas when maintained at rest exerts a tangential
force on the sides of the tube tending to move the tube in the direction of
the flow of heat, whereas if the gas were free to move it would flow towards
the hottest end ; arid this is the phenomenon of thermal transpiration.

59. The foregoing illustration, with the exception that the action is con-
fined to a plane instead of being distributed through a space, is more than
analogous : it is strictly parallel to the case of gas as long as the gas is so
rare that the molecules proceed straight across the intervals between the
plates or sides of a tube. When this is the case, therefore, the example of
the batteries explains the phenomena of thermal transpiration as well as the
phenomena of the radiometer. But when the gas is so dense that in crossing
the interval between the surfaces the molecules undergo several encounters,
the parallelism no longer holds. Even then, however, the analogy holds, for



33] IN THE GASEOUS STATE. 309

the gas at any point may be considered as consisting of two sets of molecules
which are moving across a plane from opposite sides. And by examining the
difference in the velocity of these two sets of molecules a general explanation
of many of the phenomena may be obtained without recourse being had to a
strict analytical investigation. The analogy has, however, been pursued far
enough to serve the purpose of an introduction.

Before proceeding to the mathematical investigation, which is novel and
somewhat intricate, I have thought it advisable to further introduce it by a
short description of the method used and the assumptions involved.

Prefatory description of the mathematical method.

60. The characteristic as well as the novelty of this investigation consists
in the method by which not only the mean of the motions of the molecules
at the point under consideration is taken into account, but also the manner
in which this mean motion may vary from point to point in any direction
across the point under consideration. It appears that such a variation gives
rise to certain stresses in the gas (tangential and normal), and it is of these
stresses that the phenomena of transpiration and impulsion afford evidence.

Instead of considering only the condition of the molecules comprised
within an elementary unit of volume of the gas, what is chiefly considered in
this investigation is the condition of the molecules which cross a plane sup-
posed to be drawn through the point, which plane may or may not be in
motion along its normal.

The molecules which cross this plane are considered as consisting of two
groups, one crossing from the positive to the negative side of the plane, and
the other crossing from the negative to the positive side. Considered in
opposite directions, the mean characteristics (the number, mass, velocity,
momentum, energy, &c.) of these two groups are not necessarily equal : they
may differ in consequence of the motion of the gas, the motion of the plane
through the gas, or a varying condition of the gas. And the determination
of the effects of these causes on the mass, momentum, and energy that may
be carried across by either group is the more general result of the in-
vestigation.

61. As a preliminary step, it is shown that whatever may be the nature
of the encounters between the molecules within a small element, the
encounters can produce no change on the mean component velocities of the
molecules which in a definite time pass through the element; and hence,
whatever may be the state towards which the encounters tend to reduce the
gas, this state must be such that the mean component velocities of the



310 ON CERTAIN DIMENSIONAL PROPERTIES OF MATTER [33

molecules which pass through the element in a unit of time remain unaltered.
These mean component velocities, it is to be noticed, are not the mean
component velocities of the molecules within an element at any instant.

Certain assumptions are then made. These do not involve any law of
action between the molecules. They are equivalent to assuming that the
tendency of the encounters within an element is to reduce the gas to a
uniform state.

From these assumptions two theorems (I. and II.) are deduced. From
theorem I. it follows that the rate of approximation to a uniform gas is
inversely proportional to a certain distance s, which distance is inversely
proportional to the density, and is some unknown function of the mean
velocity of the molecules. From theorem II. it follows that the molecules
which enter a small element from any particular direction, arrive as if from
the uniform gas, to which the actual gas tends at a point distant s in the
direction from which the molecules come.

When the gas is continuous about the element for distances large com-
pared with s, then s is independent of the direction from which the molecules
come ; but near a solid surface s is a function of this direction and of the
position of the element with respect to the solid surface.

These theorems are fundamental to all the reasoning which follows ; and
the distance s enters as a quantity of primary importance into all the results
obtained.

It is proposed to call this distance the mean range of the characteristics
of the molecules. Thus we have the mean range of the mass, the mean range
of momentum, and the mean range of energy. By qualifying the term
" mean range " by the name of the quantity carried, instead of considering
it as a general characteristic of the condition of the gas, two things are
avoided

(1) It is not implied that the mean range is the same for all the
quantities which may be considered;

(2) There is no fear of confusing the mean range with the mean path of
a molecule.

The mean range, whatever may be the nature of the quantity considered,
is obviously a function of the mean path of the molecules, and is a small
quantity of the same order as the mean path, but it also depends on the
nature of the impacts between the molecules.

The symbol s is used to express the mean range of any particular
quantity Q.



33] IN THE GASEOUS STATE. 311

62. Assuming that the mean value of Q for the molecules in an elementary
unit of volume at a point is a function of the position of the point, the
aggregate value of Q carried across the plane at a point is obtained in a series
of ascending powers of s. And by neglecting the terms which involve the
higher powers of s, which terms also involve differentials of Q of orders and
degrees higher than the first, equations are obtained between s and the
aggregate value of Q carried across the plane.

63. The dynamical conditions of steady momentum, steady density, and
steady pressure are next considered. General equations are obtained for
these conditions, which general equations involve s, the motion of the plane
and other quantities depending on the condition of the gas.

The condition that there may be no tangential stress in the gas is also
considered.

It is found that when there is no tangential stress on a solid surface
wherever it may be in the gas, the mean component velocities of all the
molecules which pass through the element in a definite time must be zero at
all points in the gas.

64. The equations of motion are then applied to the particular cases
which it is the object of this investigation to explain. Two cases are
considered. The first, that of a gas in which the temperature and pressure
only vary along one particular direction, so that the isothermal surfaces and
surfaces of equal pressure are parallel planes ; this is the case of transpiration.
The second case is that in which the isothermal surfaces and the surfaces of
equal pressure are curved surfaces (whether of single or double curvature) ;
this is the case of impulsion and the radiometer.

As regards the first case, the condition of steady pressure proves to be of
no importance ; but from the conditions of steady momentum and steady
density an equation is obtained between the velocity of the gas, the rate at
which the temperature varies, and the rate at which the pressure varies ; the
coefficients being functions of the absolute temperature of the gas, the
diameters of the apertures, and the ratio of the diameters of the apertures to
the mean range. These coefficients are determined in the limiting conditions
of the gas, when the density is small and large, and as they vary continuously
with the condition of the gas, the limiting values afford indications of what
must be the intermediate values.

From this equation, which is the general equation of transpiration, the
experimental results, both as regards thermal transpiration and transpiration
under pressure, are deduced.

In dealing with the second case, that in which the isothermal surfaces



312 ON CERTAIN DIMENSIONAL PROPERTIES OF MATTER [33

are curved, the three conditions steady momentum, density, and pressure
are all of them important. These conditions reduced to an equation between
the motion of the gas, the variation in the absolute temperature, and the
variation in pressure, in which, as in the equation of transpiration, the
coefficients are functions of .the absolute temperature, the diameters of the
apertures, and the ratios of the diameters of the apertures to the mean
range.

The reduction of the conditions of equilibrium to this equation, however,
involves the assumption that the gas should not be extremely rarefied. In
order to take this case into account a particular example is examined, and
the equation so obtained, together with the equation obtained from the
conditions of steady motion, is shown to lead to the results of impulsion and
the phenomena of the radiometer.



SECTION VI. NOTATION AND PRELIMINARY EXPRESSIONS.

65. In arranging the notation I have endeavoured as far as possible to
make it similar to the notation already adapted to the kinetic theory of gases
by previous writers. With this object I have adopted almost entirely, both
as regards symbols and expressions, the notation used by Professor Maxwell
in his paper "On the Dynamical Theory of Gases*." But his notation,
copious as it is, has fallen far short of my requirements. I have had to take
under consideration certain quantities which have not hitherto been
recognised ; and what has particularly taxed my resources in symbolising,
is that I have had, according to my method, to devise symbols to express
each of twenty-four partial or component quantities which spring from any
one of certain quantities, which have hitherto been dealt with as simple
quantities.

Explanation of the symbols.

66. u, v, w, are used to represent the component velocities of a molecule
with reference to the fixed axes x, y, z.

%, i), are used to represent the component velocities of a molecule with
reference to axes parallel to x, y, z, but which move with the halves of the
mean component velocities of the molecules which pass through an element
in a definite time.

U, V, W are used to represent the component velocities of the moving
axes.

* Phil. Trans. 1867.




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